#8649928

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
prime radical	#46396	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4049 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:

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

Polycyclic presentation:

2^9|a^2=g,b^2=gi,c^2=i,d^2=gi,f^2=hi,b^a=bg,c^a=ch,d^a=dg,e^a=eh,c^b=ci,d^b=dh,e^b=egi,e^c=egi,f^d=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, 8557124);