#403833

order	512
nilpotency class	3
isoclinism	isoclinic class of #395166
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁶

Aliases

No aliases for this group.

Subgroups

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

1158 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 44
transitive degree	μ_t(G) = 256

A minimal faithful permutation representation — 4 generators on 44 points:

(1,2)(3,4)(7,11)(9,12)(13,14,18,25,20,26,19,24)(15,23,17,21,29,36,28,37)(16,30,33,22,34,39,32,40)(27,35,41,38,42,44,31,43);(5,6,8,10)(7,12,11,9)(13,15,19,28,20,29,18,17)(14,21,25,36,26,37,24,23)(16,27,33,41,34,42,32,31)(22,35,30,43,40,44,39,38);(5,7,8,11)(6,9,10,12)(13,16)(14,22)(15,27)(17,31)(18,32)(19,33)(20,34)(21,35)(23,38)(24,39)(25,30)(26,40)(28,41)(29,42)(36,43)(37,44);(2,4)(13,17,18,29,20,28,19,15)(14,23,24,37,26,36,25,21)(16,31,32,42,34,41,33,27)(22,38,39,44,40,43,30,35)

Polycyclic presentation:

2^9|a^2=fgi,b^2=egi,c^2=g,d^2=e,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,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, 311029);