#412698

order	512
nilpotency class	3
isoclinism	isoclinic class of #394799
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁴×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×D₄	C₂³
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

1138 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:

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

Polycyclic presentation:

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

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, 319894);