#126532

order	512
nilpotency class	2
isoclinism	isoclinic class of #120
automorphism	?
homology	Abelianization (H₁) = C₄²×C₈ · Schur multiplier (H₂) = C₂²×C₈

Aliases

No aliases for this group.

Subgroups

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

366 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 52
transitive degree	μ_t(G) = 512

A minimal faithful permutation representation — 3 generators on 52 points:

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

Polycyclic presentation:

2^9|a^2=d,b^2=e,c^2=f,d^2=g,e^2=h,h^2=i,c^a=ci,c^b=cg

Conjugacy classes & character table

224 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 33728);