#499980

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	#24266	C₂
LCS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂×C₄×D₄ ◁ C₂×C₈×D₄ ◁ G
LECS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁

948 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)(13,14,18,22)(15,23,25,41)(16,27,31,44)(17,32,30,21)(19,34,35,24)(20,37,39,28)(26,42,33,38)(29,40,43,36);(5,7,8,12)(6,9,10,11)(15,24)(16,28)(17,30)(19,23)(20,27)(21,32)(25,34)(26,43)(29,33)(31,37)(35,41)(36,42)(38,40)(39,44);(2,4)(13,15)(14,19)(16,29)(17,33)(18,25)(20,38)(21,40)(22,35)(23,27)(24,28)(26,30)(31,43)(32,36)(34,37)(39,42)(41,44);(6,10)(9,11)(15,25)(19,35)(23,41)(24,34)(26,33)(29,43)(36,40)(38,42)

Polycyclic presentation:

2^9|a^2=i,b^2=e,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, 407176);