#8533622

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

4139 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:

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

Polycyclic presentation:

2^9|a^2=gi,b^2=i,c^2=i,d^2=gh,e^2=h,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, 8440818);