#397200

order	512
nilpotency class	3
isoclinism	isoclinic class of #394801
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

1488 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)(16,24)(17,29)(22,31)(23,33)(25,32)(27,34)(28,36)(30,35)(37,39)(38,40);(1,3,4,2)(9,11,13,24,15,23,14,25)(10,16,18,21,20,32,19,33)(12,27,28,22,30,38,29,37)(17,34,35,39,36,40,26,31);(2,3)(9,12,14,29,15,30,13,28)(10,17,19,26,20,36,18,35)(11,22,25,27,23,37,24,38)(16,31,33,40,32,39,21,34);(6,8)(11,23)(16,32)(21,33)(22,37)(24,25)(27,38)(31,39)(34,40)

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,d^b=di,g^b=gi,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, 304396);