#8926433

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

4147 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)(3,9)(4,10)(5,13)(6,8)(7,12)(11,15)(14,16)(17,18,22,20)(19,24,27,23)(21,25,29,28)(26,31,32,30)(35,41)(40,47)(42,45)(43,48);(1,3)(2,7)(4,11)(5,10)(6,12)(8,9)(13,14)(15,16)(17,19,22,27)(18,23,20,24)(21,26,29,32)(25,30,28,31)(33,34,38,39)(35,40,41,47)(36,44,46,37)(42,43,45,48);(2,8)(5,15)(7,9)(10,16)(21,29)(25,28)(26,32)(30,31)(34,39)(37,44)(40,47)(43,48);(1,3)(2,9)(4,13)(5,10)(6,12)(7,8)(11,14)(15,16)(17,20,22,18)(19,24,27,23)(21,28,29,25)(26,31,32,30)(33,35)(34,40)(36,45)(37,43)(38,41)(39,47)(42,46)(44,48);(1,4,6,14)(2,10,8,16)(3,11,12,13)(5,9,15,7)(17,19)(18,24)(20,23)(21,26)(22,27)(25,31)(28,30)(29,32)(33,36)(34,37)(35,42)(38,46)(39,44)(40,48)(41,45)(43,47);(1,5,6,15)(2,11,8,13)(3,10,12,16)(4,9,14,7)(17,21,22,29)(18,25,20,28)(19,26,27,32)(23,30,24,31)(33,37)(34,36)(35,43)(38,44)(39,46)(40,45)(41,48)(42,47)

Polycyclic presentation:

2^9|a^2=gh,b^2=ghi,d^2=gh,e^2=h,f^2=g,b^a=bg,c^a=ch,d^a=di,f^a=fh,c^b=ci,e^b=eghi,f^c=fgi,e^d=egi

Conjugacy classes & character table

80 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 8833629);