#423863

order	512
nilpotency class	3
isoclinism	isoclinic class of #417778
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁶

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	D₄²
commutator, Frattini, cosocle	C₂³×C₄	C₂⁴
prime radical	C₂²×#41	C₂³
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

1318 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

62 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 331059);