#499982

order	512
nilpotency class	4
isoclinism	isoclinic class of #499724
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁶

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	C₂²×D₈
commutator, Frattini, cosocle	C₂²×C₈	C₂⁴
prime radical	C₂²×D₄	D₈
LCS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂×C₄×D₄ ◁ C₂×C₈×D₄ ◁ G
LECS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁

928 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 44
transitive degree	μ_t(G) = 256

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

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

Polycyclic presentation:

2^9|a^2=i,b^2=e,c^2=i,d^2=i,f^2=hi,g^2=hi,h^2=i,b^a=be,c^a=cf,d^a=de,f^a=fh,g^a=gh,h^a=hi,c^b=cg,f^b=fh,g^b=gh,h^b=hi,d^c=di,f^c=fh,g^c=gh,h^c=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, 407178);