#7594358

order	512
nilpotency class	3
isoclinism	isoclinic class of #7518920
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	#24265	C₂
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2395 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1511 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)(3,5)(4,6)(7,8)(18,23)(20,29)(21,24)(25,30)(26,31)(27,32);(1,3)(2,5)(4,7)(6,8)(9,10)(11,13)(12,14)(15,16)(17,18,24,31,22,26,21,23)(19,25,32,29,28,20,27,30);(2,6)(3,7)(9,11,12,15)(10,13,14,16)(17,19,22,28)(18,25,26,20)(21,27,24,32)(23,30,31,29);(2,6)(5,8)(10,14)(13,16)(18,26)(19,28)(23,31)(27,32);(17,20,21,29,22,25,24,30)(18,27,23,28,26,32,31,19)

Polycyclic presentation:

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

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, 7501554);