#346254

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

Polycyclic presentation:

2^9|a^2=fgh,b^2=hi,c^2=ef,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	#72283940, #72284031, #72284388, …
maximal normal subgroup of	#72283940, #72284031, #72284388, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 253450);