#9712395

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

4165 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

83 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 9619591);