#124912

order	512
nilpotency class	4
isoclinism	isoclinic class of #124865
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄ · Schur multiplier (H₂) = C₂²×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₄	#1525
commutator	C₂×C₄²	C₂²×C₄
Frattini, cosocle	#376	C₂³
socle	C₂	#7637
prime radical	#3150	C₄
LCS	G ▷ C₂×C₄² ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₄ ◁ C₂²×C₈ ◁ #1726 ◁ G
LECS	G ▷ #376 ▷ C₂²×C₄ ▷ C₂ ▷ C₁

350 conjugacy classes of subgroups; only the 70 normal subgroups are shown.

normal subgroups only

This group has 350 conjugacy classes of subgroups — too many to draw. Only its 70 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)(3,13,16,37,18,39,17,38)(4,19,22,49)(5,24,26,56)(6,11)(7,12)(8,15,29,44,30,25,34,46)(9,23,32,50)(10,27,33,51)(14,40,42,36,43,60,45,61)(20,35)(21,52)(28,57,47,53,59,62,58,41)(31,55)(48,63)(54,64);(1,3)(2,8)(4,14)(5,25)(6,16)(7,18)(9,28)(10,13)(11,34)(12,30)(15,26)(17,20)(19,47)(21,53)(22,43)(23,45)(24,37)(27,44)(29,35)(31,36)(32,59)(33,39)(38,56)(40,48)(41,55)(42,50)(46,51)(49,58)(52,61)(54,62)(57,63)(60,64);(1,4,20,50,7,22,6,23)(2,9,11,19,12,32,35,49)(3,14,17,42,18,43,16,45)(5,21,51,63,26,55,27,54)(8,28,34,47,30,59,29,58)(10,31,24,48,33,52,56,64)(13,36,37,40,39,61,38,60)(15,41,44,62,25,53,46,57)

Polycyclic presentation:

2^9|a^2=e,c^2=fgh,d^2=g,e^2=h,f^2=i,b^a=bd,c^a=cfh,d^a=df,g^a=gi,d^b=dg,e^b=efgi,f^b=fi,h^b=hi,e^c=ei,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, 32108);