#412430

order	512
nilpotency class	3
isoclinism	isoclinic class of #395169
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁵×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	C₂×#121
commutator, Frattini, cosocle	C₂³×C₄	C₂⁴
prime radical	C₂²×#41	C₂³
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

1384 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 40
transitive degree	μ_t(G) = 256

A minimal faithful permutation representation — 4 generators on 40 points:

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

Polycyclic presentation:

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

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, 319626);