#474800

order	512
nilpotency class	3
isoclinism	isoclinic class of #453960
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂²×C₄

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

924 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 40
transitive degree	μ_t(G) = 512

A minimal faithful permutation representation — 4 generators on 40 points:

(3,7)(5,8)(9,10)(11,16)(12,14)(13,15)(17,18)(19,21)(20,22)(23,24)(25,26,30,38,28,31,29,33)(27,34,37,32,35,40,36,39);(1,2,4,6)(3,8,7,5)(9,11,13,12)(10,14,15,16)(17,19,20,23)(18,21,22,24)(26,31)(32,39)(33,38)(34,40);(1,3,4,7)(2,5,6,8)(10,15)(11,12)(25,27,28,35)(26,32,31,39)(29,37,30,36)(33,40,38,34);(9,12,13,11)(10,14,15,16)(18,22)(21,24)(26,31)(27,35)(33,38)(36,37)

Polycyclic presentation:

2^9|a^2=fgi,b^2=ghi,c^2=eg,d^2=ei,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, 381996);