#346256

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

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

normal subgroups only

This group has 460 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)(4,20,41,35)(5,25,26,10)(6,11)(7,13)(9,23,34,24)(12,15)(16,39,44,49)(17,46,62,36)(19,31,40,30)(21,28,45,32)(22,52,56,58)(27,38,42,37)(29,47,57,48)(33,54,50,55)(43,60)(51,64)(53,61)(59,63);(1,3,15,40)(2,8,12,31)(4,21,41,44)(5,17,42,62)(6,18,7,19)(9,32,34,39)(10,36,38,46)(11,14,13,30)(16,23,45,24)(20,49,35,28)(22,53,55,43)(25,57,37,29)(26,48,27,47)(33,61,52,60)(50,59,58,64)(51,56,63,54);(1,4,6,24)(2,9,12,34)(3,16)(5,22,27,56)(7,23,15,41)(8,28,30,32)(10,33,38,52)(11,20,13,35)(14,39,31,49)(17,43)(18,21)(19,44)(25,58,37,50)(26,54,42,55)(29,59,46,60)(36,61,57,64)(40,45)(47,51)(48,63)(53,62);(1,5,6,27)(2,10,11,37)(3,17,18,47)(4,22,24,56)(7,26,15,42)(8,29,14,36)(9,33,20,50)(12,38,13,25)(16,43,21,51)(19,48,40,62)(23,54,41,55)(28,59,39,61)(30,46,31,57)(32,60,49,64)(34,52,35,58)(44,63,45,53)

Polycyclic presentation:

2^9|a^2=fg,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	#72283625, #72283912, #72284452, …
maximal normal subgroup of	#72283625, #72283912, #72284452, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 253452);