#7510052

order	512
nilpotency class	3
isoclinism	isoclinic class of #30396
automorphism	?
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂⁷

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₂²×C₄	C₂²×D₄
commutator, Frattini, cosocle	C₂²×C₄	C₂⁵
socle	C₂³	C₂³×D₄
prime radical	C₂²×D₄	C₂×D₄
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂²×C₄ ◁ C₄²×D₄ ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1358 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 32
transitive degree	μ_t(G) = 256

A minimal faithful permutation representation — 5 generators on 32 points:

(1,2)(3,4)(5,7)(6,8)(9,10,13,15)(11,14,12,16)(17,18)(19,27)(20,25)(21,23)(22,26)(24,29)(28,31)(30,32);(1,3)(2,6)(4,7)(5,8)(9,11,13,12)(10,14,15,16)(18,23)(19,28)(20,22)(24,31)(27,32)(29,30);(9,12,13,11)(10,14,15,16)(17,19,21,29)(18,24,23,27)(20,28,22,30)(25,32,26,31);(2,7)(3,8)(10,15)(14,16)(17,20,21,22)(18,25,23,26)(19,28,29,30)(24,32,27,31);(1,4,5,6)(2,3,7,8)

Polycyclic presentation:

2^9|a^2=gi,b^2=gi,c^2=g,d^2=i,e^2=fi,h^2=i,b^a=bf,c^a=cg,d^a=dfg,c^b=ch,d^b=df,h^b=hi,h^c=hi

Conjugacy classes & character table

92 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 7417248);