#8695993

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

4193 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:

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

Polycyclic presentation:

2^9|a^2=ghi,b^2=g,c^2=i,e^2=h,f^2=g,b^a=bg,c^a=ch,d^a=dg,e^a=eh,c^b=ci,d^b=dghi,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, 8603189);