#7599994

order	512
nilpotency class	3
isoclinism	isoclinic class of #7519008
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	C₂×#121	C₂³
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2401 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1321 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:

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

Polycyclic presentation:

2^9|a^2=gi,b^2=f,c^2=gh,d^2=g,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,e^c=ei,e^d=ei

Conjugacy classes & character table

74 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 7507190);