#6998545

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

1582 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)(19,25)(20,26)(21,23)(22,24)(27,30)(28,31)(29,32);(1,2)(3,6)(4,5)(7,8)(9,10)(11,15)(12,14)(13,16)(18,23)(19,27)(24,26)(28,29);(9,11)(10,13)(12,16)(14,15)(18,24)(20,22)(23,26)(25,31)(28,29)(30,32);(2,5)(6,8)(11,16)(13,15)(17,19,21,27)(18,25,23,30)(20,29,22,28)(24,31,26,32);(1,3,4,7)(2,6,5,8)(10,14)(13,15)(17,20,21,22)(18,26,23,24)(19,28,27,29)(25,31,30,32)

Polycyclic presentation:

2^9|d^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, 6905741);