#7590180

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

1496 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)(18,23)(20,29)(21,25)(24,30)(27,32)(28,31);(1,3)(2,5)(4,7)(6,8)(9,11)(10,13)(12,15)(14,16)(17,18,21,23,22,20,25,29)(19,24,28,30,26,27,31,32);(2,6)(3,7)(17,19)(18,24)(20,27)(21,28)(22,26)(23,30)(25,31)(29,32);(2,6)(5,8)(11,15)(13,16)(19,26)(24,27)(28,31)(30,32);(17,20,21,29,22,18,25,23)(19,27,28,32,26,24,31,30)

Polycyclic presentation:

2^9|a^2=h,b^2=f,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, 7497376);