#125043

order	512
nilpotency class	4
isoclinism	isoclinic class of #124987
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄ · Schur multiplier (H₂) = C₂²×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₄	#1536
commutator	C₂×C₄²	C₂²×C₄
Frattini, cosocle	#376	C₂³
socle	C₂	#7651
prime radical	#3150	C₄
LCS	G ▷ C₂×C₄² ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₄ ◁ C₂²×C₄ ◁ #1726 ◁ G
LECS	G ▷ #376 ▷ C₂²×C₄ ▷ C₂ ▷ C₁

340 conjugacy classes of subgroups; only the 86 normal subgroups are shown.

normal subgroups only

This group has 340 conjugacy classes of subgroups — too many to draw. Only its 86 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 — 3 generators on 64 points:

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

Polycyclic presentation:

2^9|a^2=e,c^2=fgh,d^2=g,e^2=h,f^2=i,b^a=bd,c^a=cfgh,d^a=df,g^a=gi,d^b=dg,e^b=efgi,f^b=fi,h^b=hi,e^d=ei

Conjugacy classes & character table

56 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 32239);