#491661

order	512
nilpotency class	4
isoclinism	isoclinic class of #491653
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂³×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	C₂×#493
commutator, Frattini, cosocle	C₄×C₈	C₂⁴
prime radical	#33773	C₂
LCS	G ▷ C₄×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ C₂×C₈² ◁ G
LECS	G ▷ C₄×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁

531 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:

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

Polycyclic presentation:

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

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, 398857);