#8962958

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

4210 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

86 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 8870154);