#8618145

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

4158 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 48
transitive degree	μ_t(G) = 256

A minimal faithful permutation representation — 6 generators on 48 points:

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

Polycyclic presentation:

2^9|a^2=gh,b^2=hi,c^2=h,e^2=gi,f^2=g,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, 8525341);