#424816

order	512
nilpotency class	3
isoclinism	isoclinic class of #417778
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁴×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	D₄²
commutator, Frattini, cosocle	C₂³×C₄	C₂⁴
prime radical	C₂⁴	H(5,𝔽₂)
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

1215 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 44
transitive degree	μ_t(G) = 128

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

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

Polycyclic presentation:

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

Conjugacy classes & character table

62 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 332012);