#8691872

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

4175 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)(3,12)(4,10)(5,14)(6,7)(8,11)(9,13)(15,16)(17,18,23,27)(19,24,28,30)(20,25,29,32)(21,22,26,31)(33,34,38,41)(35,39,43,48)(36,44,37,40)(42,47,45,46);(1,3,8,13)(2,9,11,12)(4,7,5,15)(6,10,16,14)(19,28)(20,29)(24,30)(25,32)(33,35)(34,39)(36,45)(37,42)(38,43)(40,47)(41,48)(44,46);(1,4,8,5)(2,10,11,14)(3,7,13,15)(6,9,16,12)(17,19)(18,24)(20,21)(22,25)(23,28)(26,29)(27,30)(31,32)(33,36,38,37)(34,40,41,44)(35,42,43,45)(39,46,48,47);(1,5,8,4)(2,10,11,14)(3,7,13,15)(6,12,16,9)(17,20)(18,25)(19,21)(22,24)(23,29)(26,28)(27,32)(30,31)(33,35)(34,39)(36,42)(37,45)(38,43)(40,46)(41,48)(44,47);(1,6,8,16)(2,7,11,15)(3,10,13,14)(4,9,5,12)(17,21)(18,22)(19,20)(23,26)(24,25)(27,31)(28,29)(30,32)(33,34)(35,39)(36,40)(37,44)(38,41)(42,46)(43,48)(45,47);(1,7)(2,6)(3,5)(4,13)(8,15)(9,10)(11,16)(12,14)(17,22)(18,26)(19,25)(20,30)(21,27)(23,31)(24,29)(28,32)(33,37,38,36)(34,40,41,44)(35,42,43,45)(39,47,48,46)

Polycyclic presentation:

2^9|a^2=i,b^2=g,c^2=gh,d^2=g,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, 8599068);