#395051

order	512
nilpotency class	3
isoclinism	isoclinic class of #393944
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁶

Aliases

No aliases for this group.

Subgroups

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

1636 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 32
transitive degree	μ_t(G) = 512

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

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

Polycyclic presentation:

2^9|a^2=i,b^2=i,c^2=i,d^2=e,f^2=i,g^2=i,b^a=be,c^a=cf,d^a=dh,f^a=fi,g^a=gi,c^b=cg,f^b=fi,g^b=gi,d^c=di,f^c=fi,g^c=gi

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, 302247);