#8097282

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

4043 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:

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

Polycyclic presentation:

2^9|a^2=ghi,b^2=gh,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, 8004478);