#8097181

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

4021 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)(3,10)(4,7)(5,8)(6,9)(11,14)(12,15)(13,16)(17,18)(19,26)(20,24)(21,31)(22,27)(23,28)(25,29)(30,32)(33,34,38,41)(35,39,42,46)(36,37,40,45)(43,44,47,48);(2,4)(3,8)(9,12)(11,16)(17,19)(18,22)(20,29)(21,30)(23,27)(24,31)(25,32)(26,28)(35,42)(39,46)(43,47)(44,48);(1,3)(2,5)(4,10)(6,11)(7,8)(9,13)(12,14)(15,16)(17,20,23,30)(18,24,28,32)(19,29,27,21)(22,31,26,25)(33,35)(34,39)(36,43)(37,44)(38,42)(40,47)(41,46)(45,48);(1,4,7,2)(3,10,8,5)(6,12,15,9)(11,14,16,13)(17,21,23,29)(18,25,28,31)(19,30,27,20)(22,32,26,24);(1,5,7,10)(2,8,4,3)(6,14,15,13)(9,11,12,16)(17,19)(18,26)(20,29)(21,30)(22,28)(23,27)(24,25)(31,32)(33,36)(34,37)(35,43)(38,40)(39,44)(41,45)(42,47)(46,48);(1,6,7,15)(2,9,4,12)(3,11,8,16)(5,13,10,14)(17,22,23,26)(18,27,28,19)(20,31,30,25)(21,32,29,24)(33,37)(34,40)(35,44)(36,41)(38,45)(39,47)(42,48)(43,46)

Polycyclic presentation:

2^9|a^2=hi,c^2=gh,d^2=g,e^2=h,f^2=g,b^a=bg,c^a=ch,d^a=dg,c^b=ci,d^b=dh,f^e=fghi

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, 8004377);