#501345

order	512
nilpotency class	4
isoclinism	isoclinic class of #499712
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	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

743 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)(15,24)(20,35)(25,40)(26,34)(27,41)(31,44)(36,37)(39,42);(6,9)(10,12)(13,14)(15,25)(16,21)(17,22)(18,30)(19,33)(20,27)(23,32)(24,40)(26,42)(28,38)(29,43)(31,36)(34,39)(35,41)(37,44);(2,4)(5,7,8,11)(6,10,9,12)(13,15)(14,20)(16,26)(17,24)(18,31)(19,34)(21,36)(22,35)(23,39)(25,30)(27,32)(28,41)(29,44)(33,37)(38,42)(40,43);(6,9)(10,12)(13,16,17,19)(14,21,22,33)(15,26,24,34)(18,28,29,32)(20,36,35,37)(23,30,38,43)(25,42,40,39)(27,31,41,44)

Polycyclic presentation:

2^9|a^2=fi,c^2=e,d^2=i,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, 408541);