#509748

order	512
nilpotency class	4
isoclinism	isoclinic class of #509065
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂³×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	C₂²×D₈
commutator, Frattini, cosocle	C₂²×C₈	C₂⁴
prime radical	C₂×#2186	C₂
LCS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₄×C₂²⋊C₄ ◁ C₈×C₂²⋊C₄ ◁ G
LECS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁

745 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)(15,25)(16,22)(17,20)(18,23)(19,24)(21,28)(26,35)(27,41)(29,38)(30,34)(31,40)(32,44)(33,36)(37,42)(39,43);(5,7)(6,9)(8,12)(10,11)(14,20)(15,26)(16,19)(18,31)(21,35)(23,38)(25,37)(27,43)(28,42)(29,30)(32,33)(34,40)(36,39)(41,44);(2,4)(13,15)(14,21)(16,27)(17,28)(18,32)(19,33)(20,25)(22,36)(23,39)(24,41)(26,31)(29,42)(30,43)(34,44)(35,38)(37,40);(2,4)(6,10)(7,12)(13,16,17,19)(14,22,20,24)(15,27,28,33)(18,29,30,31)(21,36,25,41)(23,38,34,40)(26,32,42,43)(35,44,37,39)

Polycyclic presentation:

2^9|a^2=fi,d^2=i,g^2=hi,h^2=i,b^a=be,c^a=cf,d^a=def,c^b=cg,d^b=de,g^b=gh,h^b=hi,g^c=gh,h^c=hi

Conjugacy classes & character table

74 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 416944);