#7589955

order	512
nilpotency class	3
isoclinism	isoclinic class of #7514000
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×D₄²	C₂²
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₄×#123 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1870 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)(19,22)(20,23)(21,24)(25,30)(26,29)(27,31)(28,32);(2,6)(7,8)(9,11,12,16)(10,13,14,15)(17,19)(18,22)(20,26)(21,25)(23,29)(24,30)(27,28)(31,32);(1,3,5,4)(2,7,6,8)(10,14)(11,16)(18,23)(22,29)(24,32)(30,31);(1,3,5,4)(2,7,6,8)(10,14)(13,15)(19,25)(21,28)(22,30)(24,32)(26,27)(29,31);(1,4,5,3)(2,7,6,8)(10,14)(13,15)(19,26)(22,29)(25,27)(30,31)

Polycyclic presentation:

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

Conjugacy classes & character table

71 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 7497151);