#8269705

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

4134 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 48
transitive degree	μ_t(G) = 256

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

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

Polycyclic presentation:

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

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, 8176901);