#397197

order	512
nilpotency class	3
isoclinism	isoclinic class of #394799
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

1524 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:

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

Polycyclic presentation:

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

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, 304393);