#395681

order	512
nilpotency class	3
isoclinism	isoclinic class of #393904
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁶

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	C₂×#121
commutator, Frattini, cosocle	C₂³×C₄	C₂⁴
prime radical	C₂⁵	#41
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂⁴×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

1886 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)(7,8)(9,10,15,18,14,19,13,27)(11,20,29,31,22,37,23,16)(12,17,28,33,26,34,25,35)(21,36,39,30,38,32,24,40);(1,2)(3,4)(9,11,13,23,14,22,15,29)(10,16,18,20,19,31,27,37)(12,24,25,38,26,39,28,21)(17,32,33,40,34,36,35,30);(2,4)(9,12,14,26)(10,17,19,34)(11,21,22,38)(13,28,15,25)(16,30,31,40)(18,35,27,33)(20,36,37,32)(23,39,29,24);(6,8)(11,22)(16,31)(20,37)(21,38)(23,29)(24,39)(30,40)(32,36)

Polycyclic presentation:

2^9|a^2=ef,b^2=e,c^2=i,e^2=i,g^2=i,b^a=be,c^a=cf,d^a=dh,e^a=ei,f^a=fi,g^a=gi,c^b=cg,d^b=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, 302877);