#501347

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

757 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)(25,41)(26,34)(27,36)(31,44)(37,38)(40,42);(6,9)(10,12)(13,14,17,22)(15,25,24,41)(16,21,19,33)(18,30,29,43)(20,36,35,27)(23,28,39,32)(26,42,34,40)(31,37,44,38);(2,4)(5,7,8,11)(6,10,9,12)(13,15)(14,20)(16,26)(17,24)(18,31)(19,34)(21,37)(22,35)(23,40)(25,30)(27,32)(28,36)(29,44)(33,38)(39,42)(41,43);(6,9)(10,12)(13,16,17,19)(14,21,22,33)(15,26,24,34)(18,28,29,32)(20,37,35,38)(23,30,39,43)(25,42,41,40)(27,31,36,44)

Polycyclic presentation:

2^9|a^2=fi,b^2=i,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, 408543);