#444348

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

Polycyclic presentation:

2^9|a^2=e,b^2=f,c^2=g,d^2=efi,e^2=i,g^2=i,h^2=i,b^a=be,c^a=cf,d^a=dh,h^a=hi,c^b=cg,d^b=de,e^b=ei,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, 351544);