#8712323

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

4189 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:

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

Polycyclic presentation:

2^9|a^2=hi,b^2=gi,c^2=gh,d^2=gh,f^2=g,b^a=bg,c^a=ch,d^a=dg,e^a=eh,c^b=ci,d^b=dh,d^c=dgi,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, 8619519);