#412378

order	512
nilpotency class	3
isoclinism	isoclinic class of #394192
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

1400 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:

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

Polycyclic presentation:

2^9|a^2=fi,b^2=e,d^2=h,e^2=i,f^2=i,b^a=be,c^a=cf,d^a=dh,e^a=ei,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, 319574);