#405761

order	512
nilpotency class	3
isoclinism	isoclinic class of #393889
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁶

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂⁴	C₂²×D₄
commutator, Frattini, cosocle	C₂³×C₄	C₂⁴
prime radical	C₂⁵	C₂×D₄
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂⁴ ◁ C₂²×#117 ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

1954 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 32
transitive degree	μ_t(G) = 256

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

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

Polycyclic presentation:

2^9|a^2=ef,b^2=h,c^2=i,f^2=i,b^a=be,c^a=cf,d^a=dh,c^b=cg,d^b=di,f^c=fi

Conjugacy classes & character table

80 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 312957);