#489246

order	512
nilpotency class	4
isoclinism	isoclinic class of #488920
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁵

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	C₂×#447
commutator, Frattini, cosocle	C₄×C₈	C₂⁴
prime radical	#61412	C₂
LCS	G ▷ C₄×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ #2960 ◁ G
LECS	G ▷ C₄×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁

700 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 48
transitive degree	μ_t(G) = 128

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

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

Polycyclic presentation:

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

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, 396442);