#422807

order	512
nilpotency class	3
isoclinism	isoclinic class of #417464
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁶×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	C₂×#120
commutator, Frattini, cosocle	C₂³×C₄	C₂⁴
prime radical	#35634	C₂
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂⁴×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

2006 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)(9,10,14,18)(11,15,20,29)(12,16,23,32)(13,25,27,17)(19,35,36,28)(21,34,26,30)(22,31,37,40)(24,39,38,33);(1,2,3,4)(9,11)(10,15)(12,22)(13,26)(14,20)(16,31)(17,34)(18,29)(19,24)(21,27)(23,37)(25,30)(28,33)(32,40)(35,39)(36,38);(9,12)(10,16)(11,19,20,36)(13,24)(14,23)(15,28,29,35)(17,33)(18,32)(21,22,26,37)(25,39)(27,38)(30,31,34,40);(2,4)(6,8)(12,23)(16,32)(19,36)(22,37)(24,38)(28,35)(31,40)(33,39)

Polycyclic presentation:

2^9|a^2=i,b^2=e,c^2=f,g^2=i,b^a=be,c^a=cf,d^a=dh,g^a=gi,c^b=cg,d^b=de,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, 330003);