#8618112

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

4018 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:

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

Polycyclic presentation:

2^9|a^2=g,b^2=ghi,c^2=h,d^2=g,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, 8525308);