#444270

order	512
nilpotency class	3
isoclinism	isoclinic class of #418500
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	C₂³×D₄	C₂³
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

1198 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)(5,6,8,10)(7,11,12,9)(13,14)(15,20)(16,21)(17,33)(18,23)(19,25)(22,35)(24,32)(26,36)(27,42)(28,37)(29,41)(30,38)(31,39)(34,40)(43,44);(2,3)(5,7,8,12)(6,9,10,11)(13,15)(14,20)(16,31)(17,27)(18,29)(19,30)(21,39)(22,26)(23,41)(24,37)(25,38)(28,32)(33,42)(34,44)(35,36)(40,43);(6,10)(9,11)(13,16,32,40,19,33,18,22)(14,21,24,34,25,17,23,35)(15,26,29,42,30,43,28,31)(20,36,41,27,38,44,37,39);(1,3,4,2)(7,12)(9,11)(13,17,19,21)(14,22,25,40)(15,27,30,39)(16,23,33,24)(18,34,32,35)(20,26,38,43)(28,36,29,44)(31,41,42,37)

Polycyclic presentation:

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

Conjugacy classes & character table

68 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 351466);