#427233

order	512
nilpotency class	3
isoclinism	isoclinic class of #417442
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 ◁ C₂×C₄×C₂²⋊C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

1542 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:

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

Polycyclic presentation:

2^9|a^2=i,b^2=i,c^2=i,d^2=f,g^2=i,b^a=be,c^a=cf,d^a=dh,c^b=cg,d^b=de,g^b=gi,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, 334429);