#7005011

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	#45105	C₂
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2388 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1644 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:

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

Polycyclic presentation:

2^9|a^2=gi,b^2=fi,c^2=fi,g^2=i,b^a=bf,c^a=cg,d^a=di,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, 6912207);