#10195457

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

4228 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:

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

Polycyclic presentation:

2^9|a^2=gi,b^2=g,c^2=ghi,d^2=h,e^2=g,f^2=ghi,b^a=bg,c^a=ch,d^a=dg,f^a=fh,c^b=ci,e^b=eh,d^c=dh,e^d=egi,f^e=fgi

Conjugacy classes & character table

86 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 10102653);