#497068

order	512
nilpotency class	4
isoclinism	isoclinic class of #33758
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂³×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle, prime radical	C₂³	C₂²×D₈
commutator	C₂×C₈	C₂³×C₄
Frattini, cosocle	C₂²×C₈	C₂⁴
LCS	G ▷ C₂×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₄³ ◁ C₄²×C₈ ◁ G
LECS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁

568 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 44
transitive degree	μ_t(G) = 512

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

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

Polycyclic presentation:

2^9|a^2=g,b^2=gi,c^2=fh,d^2=ei,f^2=hi,h^2=i,b^a=be,c^a=cf,d^a=de,f^a=fh,h^a=hi,c^b=ce,d^c=di

Conjugacy classes & character table

80 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 404264);