#8184916

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

4153 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 48
transitive degree	μ_t(G) = 256

A minimal faithful permutation representation — 6 generators on 48 points:

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

Polycyclic presentation:

2^9|a^2=hi,b^2=ghi,c^2=gh,d^2=h,e^2=gh,f^2=h,b^a=bg,c^a=ch,d^a=dg,e^a=eh,c^b=ci,d^b=dghi,f^c=fgi

Conjugacy classes & character table

92 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 8092112);