#467139

order	512
nilpotency class	3
isoclinism	isoclinic class of #453960
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂²×C₄²

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×C₄³ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

1196 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 28
transitive degree	μ_t(G) = 512

A minimal faithful permutation representation — 4 generators on 28 points:

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

Polycyclic presentation:

2^9|a^2=fg,b^2=gi,c^2=e,d^2=e,f^2=i,b^a=be,c^a=cf,d^a=dh,c^b=cg,d^c=de,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, 374335);