#429532

order	512
nilpotency class	3
isoclinism	isoclinic class of #417603
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

1247 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

68 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 336728);