#7007879

order	512
nilpotency class	3
isoclinism	isoclinic class of #6994139
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×#447	C₂²
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2396 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1505 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:

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

Polycyclic presentation:

2^9|a^2=fgi,b^2=f,d^2=f,g^2=i,b^a=bf,c^a=cg,e^a=eh,c^b=ch,d^b=df,e^b=ehi,d^c=dh,g^c=gi,e^d=ei

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, 6915075);