#7590093

order	512
nilpotency class	3
isoclinism	isoclinic class of #7514196
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×#546	C₂²
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₄×#123 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1714 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 32
transitive degree	μ_t(G) = 128

A minimal faithful permutation representation — 5 generators on 32 points:

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

Polycyclic presentation:

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

Conjugacy classes & character table

77 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 7497289);