#8096811

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

4235 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:

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

Polycyclic presentation:

2^9|a^2=hi,c^2=g,e^2=gh,f^2=g,b^a=bg,c^a=ch,d^a=dg,c^b=ci,d^b=dh,f^e=fghi

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, 8004007);