#8696685

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

4217 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:

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

Polycyclic presentation:

2^9|a^2=ghi,b^2=g,c^2=ghi,e^2=g,f^2=h,b^a=bg,c^a=ch,d^a=dg,e^a=eh,c^b=ci,d^b=dghi,f^d=fgi,f^e=fgi

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, 8603881);