#456383

order	512
nilpotency class	3
isoclinism	isoclinic class of #453948
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁴×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	D₄²
commutator, Frattini, cosocle	C₂³×C₄	C₂⁴
prime radical	C₂×#537	C₂²
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

1217 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 40
transitive degree	μ_t(G) = 128

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

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

Polycyclic presentation:

2^9|a^2=f,b^2=f,d^2=i,e^2=i,g^2=i,b^a=be,c^a=cf,d^a=dh,e^a=ei,g^a=gi,c^b=cg,e^b=ei,g^b=gi,h^b=hi,d^c=de,e^c=ei,g^c=gi,h^c=hi,e^d=ei,g^d=gi

Conjugacy classes & character table

62 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 363579);