#8622022

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

4190 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)(3,7)(4,6)(5,8)(9,11)(10,15)(12,14)(13,16)(17,18,23,26)(19,21,24,29)(20,22,25,31)(27,28,30,32)(33,34,38,40)(35,41,43,36)(37,39,46,45)(42,47,48,44);(1,2)(3,9)(4,6)(5,8)(7,11)(10,16)(12,14)(13,15)(17,19,23,24)(18,21,26,29)(20,27,25,30)(22,28,31,32)(33,35,38,43)(34,36,40,41)(37,42,46,48)(39,44,45,47);(2,4)(7,9)(8,12)(15,16)(19,24)(21,29)(27,30)(28,32)(33,36,38,41)(34,35,40,43)(37,44,46,47)(39,42,45,48);(1,3)(2,7)(4,9)(5,13)(6,11)(8,16)(10,14)(12,15)(17,20)(18,22)(19,27)(21,28)(23,25)(24,30)(26,31)(29,32)(33,34)(35,36)(37,45)(38,40)(39,46)(41,43)(42,47)(44,48);(1,4,6,2)(3,9,11,7)(5,12,14,8)(10,16,13,15)(17,21,23,29)(18,24,26,19)(20,28,25,32)(22,30,31,27)(33,36)(34,35)(37,44)(38,41)(39,42)(40,43)(45,48)(46,47);(1,5,6,14)(2,8,4,12)(3,10,11,13)(7,15,9,16)(17,22)(18,25)(19,28)(20,26)(21,30)(23,31)(24,32)(27,29)(33,37)(34,39)(35,42)(36,44)(38,46)(40,45)(41,47)(43,48)

Polycyclic presentation:

2^9|a^2=ghi,b^2=ghi,c^2=g,e^2=i,f^2=h,b^a=bg,c^a=ch,d^a=dg,e^a=eh,c^b=ci,d^b=dh,e^b=egi,e^c=egi,f^d=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, 8529218);