#439545

order	512
nilpotency class	3
isoclinism	isoclinic class of #417778
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂²×C₄²

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

1121 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)(15,20)(16,21)(17,32)(18,23)(19,25)(22,30)(24,34)(26,35)(27,39)(28,36)(29,38)(31,41)(33,44)(37,43)(40,42);(2,3)(5,7,9,8)(6,10,11,12)(13,15,18,28,19,29,34,43)(14,20,23,36,25,38,24,37)(16,27,30,42,17,33,31,26)(21,39,22,40,32,44,41,35);(1,2,4,3)(5,8,9,7)(6,10,11,12)(13,16,19,17)(14,21,25,32)(15,26,29,42)(18,31,34,30)(20,35,38,40)(22,23,41,24)(27,28,33,43)(36,44,37,39);(1,3,4,2)(13,17,19,16)(14,22,25,41)(15,27,29,33)(18,31,34,30)(20,35,38,40)(21,23,32,24)(26,28,42,43)(36,39,37,44)

Polycyclic presentation:

2^9|a^2=ef,b^2=efg,c^2=efi,d^2=ei,g^2=i,h^2=i,b^a=be,c^a=cf,d^a=dh,h^a=hi,c^b=cg,d^b=dei,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, 346741);