#8193729

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) = 128

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

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

Polycyclic presentation:

2^9|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, 8100925);