#8621290

order	512
nilpotency class	2
automorphism	?
homology	Abelianization (H₁) = C₂⁶ · Schur multiplier (H₂) = C₂¹²

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, commutator, Frattini, socle, cosocle	C₂³	C₂⁶
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4177 conjugacy classes of subgroups.

Full subgroup data →

Presentation

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

A minimal faithful permutation representation — 6 generators on 48 points:

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

Polycyclic presentation:

2^9|a^2=ghi,b^2=gi,e^2=hi,f^2=g,b^a=bg,c^a=ch,d^a=dg,e^a=eh,c^b=ci,d^b=dh,e^b=egi,e^c=egi,f^d=fgi

Conjugacy classes & character table

89 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 8528486);