#499248

order	512
nilpotency class	4
isoclinism	isoclinic class of #498791
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁵

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	C₂²×D₈
commutator, Frattini, cosocle	C₂²×C₈	C₂⁴
prime radical	C₂×#3176	C₂
LCS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂×#117 ◁ C₂×#405 ◁ G
LECS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁

881 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)(7,10)(8,9)(11,12)(13,14)(15,21)(16,28)(17,23)(18,32)(19,25)(20,36)(22,30)(24,35)(26,37)(27,42)(29,39)(31,43)(33,41)(34,40)(38,44);(1,2)(3,4)(6,9)(10,12)(13,15)(14,21)(16,19)(17,29)(18,27)(20,26)(22,40)(23,39)(24,38)(25,28)(30,34)(31,33)(32,42)(35,44)(36,37)(41,43);(2,4)(13,16,19,15)(14,22,25,42)(17,29,20,26)(18,33,34,44)(21,32,28,40)(23,41,36,38)(24,37,43,39)(27,35,30,31);(5,7,8,11)(6,10,9,12)(13,17,19,20)(14,23,25,36)(15,26,16,29)(18,31,34,35)(21,37,28,39)(22,41,42,38)(24,32,43,40)(27,44,30,33)

Polycyclic presentation:

2^9|c^2=i,d^2=ei,f^2=hi,h^2=i,b^a=be,c^a=cf,f^a=fh,h^a=hi,c^b=cg,d^b=di,f^c=fh,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, 406444);