#7617294

order	512
nilpotency class	3
isoclinism	isoclinic class of #7518936
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 ◁ C₄×C₄⋊Q₈ ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1270 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

80 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 7524490);