#458591

order	512
nilpotency class	3
isoclinism	isoclinic class of #453981
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂²×C₄²

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

1128 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)(5,6)(7,8)(9,10,13,15)(11,14,12,16)(17,18,21,25)(19,27,30,23)(20,24,22,26)(28,32,29,31);(6,7)(9,11,13,12)(10,14,15,16)(17,19,21,30)(18,23,25,27)(20,29,22,28)(24,32,26,31);(5,7,8,6)(9,12,13,11)(10,14,15,16)(19,28)(20,22)(23,31)(24,26)(27,32)(29,30);(2,4)(6,7)(17,20,21,22)(18,24,25,26)(19,29,30,28)(23,32,27,31)

Polycyclic presentation:

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

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, 365787);