#412120

order	512
nilpotency class	3
isoclinism	isoclinic class of #394189
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	C₂³×D₈	C₂²
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

1600 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

80 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 319316);