#126538

order	512
nilpotency class	2
isoclinism	isoclinic class of #7855
automorphism	?
homology	Abelianization (H₁) = C₄³ · Schur multiplier (H₂) = C₂²×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₂²×C₄	C₂×C₄²
commutator	C₂×C₄	C₄³
Frattini, cosocle	C₂×C₄×C₈	C₂³
socle, prime radical	C₂³	C₂×#97
LCS	G ▷ C₂×C₄ ▷ C₁
UCS	C₁ ◁ C₂²×C₄ ◁ G
LECS	G ▷ C₂×C₄×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁

314 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)(23,31,32,43)(24,35)(25,30)(26,28)(27,34,39,46)(29,37)(33,47,45,44)(36,50)(38,41)(40,52,49,48)(42,51);(5,7)(6,10)(8,14)(9,13)(11,17)(12,16)(15,19)(18,20)(21,23,24,33,25,34,36,48,26,32,37,45,38,46,51,52)(22,27,29,40,30,31,42,44,28,39,35,49,41,43,50,47);(1,2,3,4)(7,13)(10,16)(14,19)(17,20)(22,28)(27,39)(29,35)(30,41)(31,43)(40,49)(42,50)(44,47)

Polycyclic presentation:

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

Conjugacy classes & character table

128 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 33734);