#8096855

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

4203 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:

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

Polycyclic presentation:

2^9|a^2=ghi,b^2=g,c^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, 8004051);