#6489040

order	512
nilpotency class	3
isoclinism	isoclinic class of #6469971
automorphism	?
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂⁶×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	C₂³×D₄
commutator, Frattini, cosocle	C₂²×C₄	C₂⁵
prime radical	C₂³×D₄	C₂³
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₄×#117 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1704 conjugacy classes of subgroups.

Full subgroup data →

Presentation

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

A minimal faithful permutation representation — 5 generators on 48 points:

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

Polycyclic presentation:

2^9|a^2=i,c^2=ghi,d^2=hi,e^2=gi,f^2=i,b^a=bf,c^a=cg,e^a=eh,f^a=fi,c^b=ch,d^b=dh,f^b=fi,e^c=ei

Conjugacy classes & character table

74 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6396236);