#490619

order	512
nilpotency class	4
isoclinism	isoclinic class of #490479
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁴

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	C₂×#466
commutator, Frattini, cosocle	C₄×C₈	C₂⁴
prime radical	C₂³	#466
LCS	G ▷ C₄×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ #2960 ◁ G
LECS	G ▷ C₄×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁

397 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 48
transitive degree	μ_t(G) = 256

A minimal faithful permutation representation — 4 generators on 48 points:

(1,2,5,8)(3,9,11,16)(4,12,14,7)(6,13,15,10)(18,24)(19,30)(20,25)(21,22)(26,41)(27,37)(28,39)(29,38)(31,35)(32,45)(33,34)(36,42)(43,48)(44,47);(17,18,25,39,22,29,34,37,23,27,33,38,21,28,20,24)(19,26,40,41,30,42,45,47,31,43,46,48,35,44,32,36);(1,3,5,11)(2,6,8,15)(4,13,14,10)(7,16,12,9)(17,19,23,31)(18,26,27,43)(20,32,34,45)(21,35,22,30)(24,36,37,47)(25,40,33,46)(28,44,29,42)(38,48,39,41);(18,27)(19,31)(24,37)(28,29)(30,35)(32,45)(38,39)(40,46)

Polycyclic presentation:

2^9|a^2=hi,b^2=eg,c^2=h,e^2=gi,f^2=h,g^2=i,b^a=be,c^a=cf,e^a=eg,f^a=fh,g^a=gi,d^b=di,d^c=di,f^c=fh

Conjugacy classes & character table

68 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 397815);