#501476

order	512
nilpotency class	4
isoclinism	isoclinic class of #499712
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	#13325	C₂
LCS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₄×C₄⋊C₄ ◁ C₈×C₄⋊C₄ ◁ G
LECS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁

679 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)(8,9)(10,12)(13,14)(15,23)(16,20)(17,21)(18,22)(19,25)(24,39)(26,40)(27,38)(28,35)(29,36)(30,37)(31,34)(32,41)(33,42)(43,44);(6,9)(10,11)(13,15)(14,19)(16,25)(17,24)(18,31)(20,23)(21,32)(22,38)(26,30)(27,29)(28,41)(33,37)(34,36)(35,39)(40,44)(42,43);(2,4)(15,24)(17,30)(18,29)(19,32)(21,37)(22,36)(23,39)(25,41)(26,27)(28,43)(31,42)(33,34)(35,44)(38,40);(5,7,8,12)(6,10,9,11)(14,20)(19,23)(21,35)(22,36)(32,39)(33,40)(34,38)(37,44)

Polycyclic presentation:

2^9|a^2=f,d^2=ei,g^2=hi,h^2=i,b^a=be,c^a=cf,d^a=de,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, 408672);