#7605828

order	512
nilpotency class	3
isoclinism	isoclinic class of #7518944
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

1209 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

68 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 7513024);