#7620689

order	512
nilpotency class	3
isoclinism	isoclinic class of #7518904
automorphism	?
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂⁷

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	C₂³×D₄
commutator, Frattini, cosocle	C₂²×C₄	C₂⁵
prime radical	#24266	C₂
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₄×#124 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1552 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)(3,7)(4,6)(5,8)(10,14)(11,12)(17,18,21,25)(19,26,22,23)(20,28,30,24)(27,31,29,32);(1,3,5,4)(2,6,8,7)(9,10,13,14)(11,16,12,15)(19,22)(20,29)(23,26)(24,31)(27,30)(28,32);(1,3,5,4)(2,7,8,6)(9,11,13,12)(10,15,14,16)(17,19,21,22)(18,23,25,26)(20,27,30,29)(24,31,28,32);(1,4,5,3)(2,6,8,7)(9,11,13,12)(10,15,14,16)(17,20,19,27,21,30,22,29)(18,24,26,32,25,28,23,31);(2,8)(6,7)(9,12,13,11)(10,15,14,16)

Polycyclic presentation:

2^9|a^2=i,b^2=f,c^2=fi,d^2=fgh,e^2=gi,h^2=i,b^a=bf,c^a=cg,d^a=dg,e^a=ef,c^b=cg,d^b=dh,h^b=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, 7527885);