#8696261

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

4182 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:

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

Polycyclic presentation:

2^9|a^2=gi,c^2=i,d^2=gh,e^2=h,f^2=g,b^a=bg,c^a=ch,d^a=dg,e^a=eh,c^b=ci,d^b=dghi,f^d=fgi,f^e=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, 8603457);