#500612

order	512
nilpotency class	4
isoclinism	isoclinic class of #499712
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	#13325	C₂
LCS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂×C₄×Q₈ ◁ C₂×C₈×Q₈ ◁ G
LECS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁

870 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)(15,24)(16,21)(17,20)(18,32)(19,26)(22,31)(23,30)(25,40)(27,43)(28,35)(29,36)(33,39)(34,41)(37,44)(38,42);(6,9)(10,11)(13,15)(14,19)(16,26)(17,25)(18,33)(20,34)(21,24)(22,38)(23,39)(27,31)(28,30)(29,41)(32,35)(36,40)(37,43)(42,44);(2,4)(14,20)(15,25)(17,31)(18,30)(21,36)(22,23)(26,41)(27,28)(29,44)(32,37)(33,42)(34,38)(35,39)(40,43);(5,7,8,12)(6,10,9,11)(14,21)(19,24)(20,36)(22,37)(23,32)(34,40)(35,39)(38,43)

Polycyclic presentation:

2^9|d^2=ei,f^2=hi,g^2=hi,h^2=i,b^a=be,c^a=cf,d^a=de,f^a=fh,g^a=gh,h^a=hi,c^b=cg,f^b=fh,g^b=gh,h^b=hi,f^c=fh,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, 407808);