#8522134

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

4125 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:

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

Polycyclic presentation:

2^9|a^2=gi,b^2=i,c^2=gh,d^2=gh,e^2=g,f^2=g,b^a=bg,c^a=ch,d^a=dg,e^a=eh,c^b=ci,d^b=dh,e^c=egi,f^d=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, 8429330);