#499409

order	512
nilpotency class	4
isoclinism	isoclinic class of #498791
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁴

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	C₂²×D₈
commutator, Frattini, cosocle	C₂²×C₈	C₂⁴
prime radical	C₂⁴	C₂×D₈
LCS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂×#117 ◁ C₂×#405 ◁ G
LECS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁

632 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)(7,12)(8,10)(9,11)(13,14)(15,19)(16,21)(17,22)(18,20)(23,31)(24,33)(25,34)(26,32)(27,37)(28,36)(29,35)(30,38)(39,43)(40,42)(41,44);(5,7,9,8)(6,10,11,12)(13,15,18,26)(14,19,20,32)(16,23,29,40)(17,30,28,25)(21,31,35,42)(22,38,36,34)(24,41,39,27)(33,44,43,37);(2,4)(5,8,9,7)(6,10,11,12)(15,23)(16,27)(17,28)(19,31)(21,37)(22,36)(24,25)(26,40)(29,41)(30,39)(32,42)(33,34)(35,44)(38,43);(14,20)(19,32)(21,35)(22,36)(31,42)(33,43)(34,38)(37,44)

Polycyclic presentation:

2^9|a^2=f,b^2=ei,c^2=e,g^2=hi,h^2=i,b^a=be,c^a=cf,d^a=di,c^b=cg,g^b=gh,h^b=hi,g^c=gh,h^c=hi

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, 406605);