#7617530

order	512
nilpotency class	3
isoclinism	isoclinic class of #7519008
automorphism	?
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂⁷

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle, prime radical	C₂³	C₂³×D₄
commutator, Frattini, cosocle	C₂²×C₄	C₂⁵
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2402 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1222 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 32
transitive degree	μ_t(G) = 512

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

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

Polycyclic presentation:

2^9|a^2=gi,b^2=gi,c^2=fgi,d^2=hi,e^2=fi,h^2=i,b^a=bf,c^a=cg,d^a=dg,e^a=ef,c^b=cg,d^b=dh,h^b=hi,e^c=ei

Conjugacy classes & character table

74 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 7524726);