#8751976

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

4197 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:

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

Polycyclic presentation:

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

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, 8659172);