#346264

order	512
nilpotency class	3
isoclinism	isoclinic class of #346224
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂³

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#2760
commutator, Frattini, prime radical, cosocle	C₂⁵	C₂⁴
LCS	G ▷ C₂⁵ ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂⁵ ◁ G
LECS	G ▷ C₂⁵ ▷ C₂² ▷ C₁

456 conjugacy classes of subgroups; only the 92 normal subgroups are shown.

normal subgroups only

This group has 456 conjugacy classes of subgroups — too many to draw. Only its 92 normal subgroups are shown.

Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.

Full subgroup data →

Presentation

minimal degree	μ(G) = 64
transitive degree	μ_t(G) = 64

A minimal faithful permutation representation — 4 generators on 64 points:

(1,2)(3,15,18,9)(4,21,29,37)(5,26,28,11)(6,13)(7,12)(8,14)(10,24,36,25)(16,34,22,50)(17,45,48,38)(19,33,20,41)(23,52,54,56)(27,40,30,39)(31,43,59,44)(32,46,55,47)(35,53,49,57)(42,61)(51,58)(60,62)(63,64);(1,3,6,19)(2,9,13,41)(4,22,29,44)(5,17,30,48)(7,18,8,20)(10,34,36,59)(11,38,40,45)(12,15,14,33)(16,25,43,24)(21,31,37,50)(23,42,57,62)(26,55,39,32)(27,46,28,47)(35,61,52,60)(49,64,56,58)(51,54,63,53);(1,4,7,25)(2,10,13,36)(3,16)(5,23,27,54)(6,29,8,24)(9,31,33,59)(11,35,40,52)(12,21,14,37)(15,34,41,50)(17,42)(18,44)(19,43)(20,22)(26,56,39,49)(28,53,30,57)(32,58,45,61)(38,60,55,64)(46,63)(47,51)(48,62);(1,5,7,27)(2,11,12,39)(3,17,18,46)(4,23,25,54)(6,30,8,28)(9,32,15,38)(10,35,21,49)(13,40,14,26)(16,42,44,63)(19,48,20,47)(22,51,43,62)(24,53,29,57)(31,58,34,60)(33,45,41,55)(36,52,37,56)(50,64,59,61)

Polycyclic presentation:

2^9|a^2=fghi,b^2=hi,c^2=efh,d^2=h,b^a=be,c^a=cf,d^a=def,e^a=eh,f^a=fhi,g^a=ghi,c^b=cg,d^b=de,f^b=fh,e^c=ei,f^c=fi,g^c=gh,g^d=gi

Conjugacy classes & character table

38 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72283608, #72283815, #72283824, …
maximal normal subgroup of	#72283608, #72283815, #72283824, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 253460);