#506495

order	512
nilpotency class	4
isoclinism	isoclinic class of #501064
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 ◁ C₂×#117 ◁ C₂×#405 ◁ G
LECS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁

819 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:

(5,6)(7,11)(8,10)(9,12)(13,14,18,24,16,29,31,36,17,23,32,43,19,22,34,42)(15,25,28,40,30,39,44,41,27,20,35,38,26,37,33,21);(1,2,3,4)(5,7,8,12)(6,9,10,11)(13,15)(14,20)(16,30)(17,27)(18,28)(19,26)(21,36)(22,39)(23,25)(24,38)(29,37)(31,44)(32,35)(33,34)(40,43)(41,42);(2,4)(13,15)(14,21)(16,26)(17,27)(18,33)(19,30)(20,36)(22,40)(23,41)(24,37)(25,42)(28,34)(29,38)(31,35)(32,44)(39,43);(6,10)(9,11)(13,16,17,19)(14,22,23,29)(15,26,27,30)(18,31,32,34)(20,37,25,39)(21,40,41,38)(24,42,43,36)(28,33,35,44)

Polycyclic presentation:

2^9|a^2=f,b^2=egi,d^2=i,f^2=hi,h^2=i,b^a=be,c^a=cf,d^a=de,c^b=cg,d^b=di,f^c=fh,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, 413691);