#508168

order	512
nilpotency class	4
isoclinism	isoclinic class of #508008
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁴×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×#466	C₂²
LCS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂×#117 ◁ C₂×#405 ◁ G
LECS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁

847 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)(13,14,16,21)(15,24,25,19)(17,30,28,35)(18,33,29,23)(20,38,39,26)(22,41,40,32)(27,37,34,42)(31,43,44,36);(1,2)(3,4)(6,9)(10,12)(13,15)(14,19)(16,25)(17,26)(18,27)(20,35)(21,24)(22,36)(23,37)(28,38)(29,34)(30,39)(31,32)(33,42)(40,43)(41,44);(2,4)(7,11)(10,12)(13,15)(14,20)(16,25)(17,31)(18,34)(19,35)(21,39)(22,42)(23,43)(24,30)(26,32)(27,29)(28,44)(33,36)(37,40)(38,41);(5,7,8,11)(6,10,9,12)(15,25)(19,24)(20,39)(26,38)(27,34)(31,44)(36,43)(37,42)

Polycyclic presentation:

2^9|a^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,d^c=de,f^c=fh,h^c=hi

Conjugacy classes & character table

62 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 415364);