#415591

order	512
nilpotency class	3
isoclinism	isoclinic class of #393877
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁵

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	C₂×#120
commutator, Frattini, cosocle	C₂³×C₄	C₂⁴
prime radical	C₂⁴	C₂²×D₄
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

1130 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 32
transitive degree	μ_t(G) = 128

A minimal faithful permutation representation — 4 generators on 32 points:

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

Polycyclic presentation:

2^9|a^2=eh,b^2=g,c^2=g,d^2=hi,e^2=i,f^2=i,b^a=be,c^a=cf,d^a=dh,f^a=fi,c^b=cg,e^b=ei,f^c=fi

Conjugacy classes & character table

74 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 322787);