#8184863

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

4157 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)(3,9,11,5)(4,12,14,8)(10,13,16,15)(17,18)(19,21)(20,23)(22,30)(24,26)(25,32)(27,28)(29,31)(33,34)(35,41)(36,37)(38,42)(39,40)(43,48)(44,45)(46,47);(1,3)(2,5)(4,10)(6,11)(7,9)(8,13)(12,15)(14,16)(17,19,24,23)(18,21,26,20)(22,27,32,31)(25,29,30,28)(34,40)(36,41)(42,47)(44,48);(2,7)(5,9)(8,12)(13,15)(17,20,24,21)(18,19,26,23)(22,28,32,29)(25,27,30,31)(33,35,39,37)(34,41,40,36)(38,45,46,43)(42,44,47,48);(1,4,6,14)(2,8,7,12)(3,10,11,16)(5,13,9,15)(17,21,24,20)(18,19,26,23)(22,29,32,28)(25,27,30,31)(33,36)(34,35)(37,40)(38,44)(39,41)(42,43)(45,47)(46,48);(1,3)(2,5)(4,10)(6,11)(7,9)(8,13)(12,15)(14,16)(17,22,24,32)(18,25,26,30)(19,27,23,31)(20,28,21,29)(33,37,39,35)(34,36,40,41)(38,45,46,43)(42,44,47,48);(1,5)(2,3)(4,13)(6,9)(7,11)(8,10)(12,16)(14,15)(17,23,24,19)(18,20,26,21)(22,31,32,27)(25,28,30,29)(33,38,39,46)(34,42,40,47)(35,43,37,45)(36,44,41,48)

Polycyclic presentation:

2^9|a^2=hi,b^2=i,c^2=gh,d^2=h,e^2=gh,f^2=gh,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, 8092059);