#125059

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

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	#7652
prime radical	C₂³	#421
LCS	G ▷ C₂²×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₄ ◁ C₂²×C₄ ◁ #1726 ◁ G
LECS	G ▷ C₂×#97 ▷ C₂²×C₄ ▷ C₂ ▷ C₁

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

normal subgroups only

This group has 251 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)(3,13,16,39,17,40,18,33)(4,19)(5,10,25,34)(6,12)(7,11)(8,15,28,24,29,47,36,48)(9,30)(14,41,43,61,44,58,45,38)(20,37)(21,50,53,64)(22,51)(23,32)(26,35,46,60)(27,52,55,62,56,63,49,42)(31,54,59,57);(1,3,6,17)(2,8)(4,14,22,44)(5,24,25,48)(7,16,20,18)(9,27)(10,33)(11,36)(12,29)(13,35)(15,46,47,26)(19,49)(21,52,53,63)(23,43,30,45)(28,37)(31,58)(32,56)(34,39)(38,50)(40,60)(41,59)(42,57,62,54)(51,55)(61,64);(1,4,20,30,6,22,7,23)(2,9,11,19,12,32,37,51)(3,14,18,45,17,44,16,43)(5,21,46,57,25,53,26,54)(8,27,36,49,29,56,28,55)(10,31,35,50,34,59,60,64)(13,38,39,41,40,61,33,58)(15,42,24,52,47,62,48,63)

Polycyclic presentation:

2^9|a^2=e,b^2=f,c^2=fgh,d^2=g,e^2=h,g^2=i,b^a=bd,c^a=cfgh,d^a=df,f^a=fi,d^b=dg,e^b=efgi,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, 32255);