#411693

order	512
nilpotency class	3
isoclinism	isoclinic class of #395242
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁷

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	D₄²
commutator, Frattini, cosocle	C₂³×C₄	C₂⁴
prime radical	C₂²×#135	C₂²
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

1834 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)(9,10,16,22)(11,23,27,35)(12,18,30,34)(13,31,33,40)(14,21,15,20)(17,26,36,25)(19,24,37,32)(28,39,29,38);(2,4)(9,11)(10,17)(12,13)(14,26)(15,25)(16,27)(18,37)(19,34)(20,35)(21,23)(22,36)(24,29)(28,32)(30,33)(31,39)(38,40);(6,8)(9,12)(10,18)(11,24)(13,25)(14,29)(15,28)(16,30)(17,31)(19,35)(20,39)(21,38)(22,34)(23,37)(26,33)(27,32)(36,40);(9,13,14,24,16,33,15,32)(10,19,20,40,22,37,21,31)(11,12,26,29,27,30,25,28)(17,34,35,38,36,18,23,39)

Polycyclic presentation:

2^9|a^2=i,d^2=hi,e^2=i,g^2=i,h^2=i,b^a=be,c^a=cf,d^a=dh,e^a=ei,g^a=gi,h^a=hi,c^b=cg,e^b=ei,g^b=gi,d^c=di,e^c=ei,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, 318889);