#8533248

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

4245 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:

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

Polycyclic presentation:

2^9|a^2=ghi,b^2=hi,c^2=gi,d^2=g,e^2=g,f^2=gh,b^a=bg,c^a=ch,d^a=dg,e^a=eh,c^b=ci,d^b=dh,e^c=egi,f^d=fgi

Conjugacy classes & character table

83 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 8440444);