#6998563

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	C₂²×D₄	C₂×D₄
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2395 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1467 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:

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

Polycyclic presentation:

2^9|a^2=i,c^2=i,e^2=f,g^2=i,b^a=bf,c^a=cg,e^a=eh,g^a=gi,c^b=ch,d^b=df,e^b=eh,d^c=dh,e^c=ei,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, 6905759);