#499464

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	C₂²×D₁₆	C₂²
LCS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #376 ◁ #1726 ◁ G
LECS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁

644 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)(7,10)(8,9)(11,12)(13,14)(15,19)(16,20)(17,21)(18,22)(23,37)(24,32)(25,33)(26,34)(27,35)(28,36)(29,39)(30,40)(31,41)(38,43)(42,44);(6,9)(10,12)(13,15)(14,19)(16,24)(17,23)(18,30)(20,32)(21,37)(22,40)(25,29)(26,28)(27,41)(31,35)(33,39)(34,36)(38,44)(42,43);(2,4)(14,20)(15,23)(17,29)(18,28)(19,31)(21,38)(24,41)(25,26)(27,43)(30,42)(32,37)(33,40)(34,44)(35,39);(5,7,8,11)(6,10,9,12)(14,20)(19,32)(21,35)(22,36)(31,37)(33,44)(34,40)(38,39)

Polycyclic presentation:

2^9|a^2=fi,d^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, 406660);