#8193730

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

4138 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:

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

Polycyclic presentation:

2^9|a^2=g,b^2=h,c^2=g,d^2=ghi,e^2=ghi,f^2=g,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, 8100926);