#8717979

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

4197 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)(3,10)(4,12)(5,7)(6,9)(8,13)(11,15)(14,16)(17,18,23,25)(19,27,21,24)(20,30,31,26)(22,28,32,29)(33,34,39,43)(35,40,44,47)(36,41,45,46)(37,38,42,48);(1,3,6,5)(2,7,9,10)(4,11,13,14)(8,15,12,16)(17,19,23,21)(18,24,25,27)(20,28,31,29)(22,26,32,30)(33,35,39,44)(34,40,43,47)(36,37,45,42)(38,46,48,41);(4,13)(8,12)(11,14)(15,16)(18,25)(20,31)(24,27)(28,29)(35,44)(36,45)(40,47)(41,46);(1,4,6,13)(2,8,9,12)(3,11,5,14)(7,15,10,16)(17,20,23,31)(18,26,25,30)(19,29,21,28)(22,27,32,24)(33,36,39,45)(34,41,43,46)(35,37,44,42)(38,47,48,40);(1,5,6,3)(2,7,9,10)(4,14,13,11)(8,15,12,16)(17,21,23,19)(18,27,25,24)(20,28,31,29)(22,26,32,30)(33,37)(34,38)(35,45)(36,44)(39,42)(40,46)(41,47)(43,48);(1,2)(3,7)(4,12)(5,10)(6,9)(8,13)(11,16)(14,15)(17,22,23,32)(18,28,25,29)(19,26,21,30)(20,24,31,27)(33,38)(34,42)(35,46)(36,40)(37,43)(39,48)(41,44)(45,47)

Polycyclic presentation:

2^9|a^2=hi,b^2=gi,d^2=gi,e^2=g,f^2=h,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, 8625175);