#7007829

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×#121	C₂³
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2396 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1378 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:

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

Polycyclic presentation:

2^9|a^2=fgi,b^2=fi,c^2=i,d^2=fi,g^2=i,b^a=bf,c^a=cg,e^a=eh,c^b=ch,d^b=df,e^b=eh,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, 6915025);