#8963651

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

4045 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:

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

Polycyclic presentation:

2^9|a^2=g,b^2=hi,c^2=h,d^2=gi,e^2=i,f^2=g,b^a=bg,c^a=ch,d^a=dg,f^a=fh,c^b=ci,e^b=eh,e^c=egi,f^c=fgi,e^d=egi

Conjugacy classes & character table

86 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 8870847);