#7590193

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	#2388	C₂²
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2388 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1487 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)(9,10,12,14)(11,13,15,16)(17,18,24,20)(19,27,30,32)(21,26,23,29)(22,28,25,31);(1,3)(2,5)(4,7)(6,8)(9,11)(10,13)(12,15)(14,16)(17,19,23,28,24,30,21,31)(18,25,26,27,20,22,29,32);(2,6)(3,7)(17,20,24,18)(19,22,30,25)(21,26,23,29)(27,28,32,31);(2,6)(5,8)(11,15)(13,16)(17,21,24,23)(18,26,20,29)(19,28,30,31)(22,27,25,32);(17,22,21,27,24,25,23,32)(18,19,29,31,20,30,26,28)

Polycyclic presentation:

2^9|a^2=h,b^2=f,c^2=i,d^2=i,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,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, 7497389);