#125044

order	512
nilpotency class	4
isoclinism	isoclinic class of #124974
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄ · Schur multiplier (H₂) = C₂³

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₄	#1517
commutator	C₂²×C₈	C₂²×C₄
Frattini, cosocle	C₂×#97	C₂³
socle	C₂	#7651
prime radical	#3150	C₄
LCS	G ▷ C₂²×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₄ ◁ C₂²×C₄ ◁ #1726 ◁ G
LECS	G ▷ C₂×#97 ▷ C₂²×C₄ ▷ C₂ ▷ C₁

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

normal subgroups only

This group has 337 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)(3,13,16,34,17,39,26,54)(4,18)(5,10,24,36)(6,12)(7,11)(8,15,28,44,30,45,29,23)(9,31)(14,40,42,38,43,58,53,61)(19,35)(20,47,51,64)(21,48)(22,33)(25,37,49,60)(27,50,46,41,56,62,55,63)(32,52,59,57);(1,3)(2,8)(4,14)(5,23)(6,17)(7,26)(9,27)(10,34)(11,28)(12,30)(13,37)(15,25)(16,19)(18,46)(20,50)(21,43)(22,53)(24,44)(29,35)(31,42)(32,58)(33,56)(36,54)(38,47)(39,60)(40,59)(41,52)(45,49)(48,55)(51,62)(57,63)(61,64);(1,4,19,31,6,21,7,22)(2,9,11,18,12,33,35,48)(3,14,16,42,17,43,26,53)(5,20,49,57,24,51,25,52)(8,27,28,46,30,56,29,55)(10,32,37,47,36,59,60,64)(13,38,54,40,39,61,34,58)(15,41,23,50,45,63,44,62)

Polycyclic presentation:

2^9|a^2=e,c^2=fgh,d^2=gi,e^2=h,g^2=i,b^a=bd,c^a=cfgh,d^a=df,f^a=fi,d^b=dg,e^b=efg,g^b=gi,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, 32240);