#458692

order	512
nilpotency class	3
isoclinism	isoclinic class of #453948
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂³×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	D₄²
commutator, Frattini, cosocle	C₂³×C₄	C₂⁴
prime radical	C₂²×#41	C₂³
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

1034 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:

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

Polycyclic presentation:

2^9|a^2=ef,b^2=fi,c^2=efi,d^2=i,g^2=i,h^2=i,b^a=be,c^a=cf,d^a=dh,h^a=hi,c^b=cg,g^b=gi,d^c=de,g^c=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, 365888);