#10412326

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

4273 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)(3,11,14,8)(4,5,10,9)(12,16,15,13)(17,18,23,26)(19,24,30,32)(20,25,28,29)(21,31,27,22)(33,34,40,44)(35,45,48,41)(36,37,38,42)(39,46,47,43);(2,7)(4,10)(8,11)(12,15)(17,19)(18,24)(20,21)(22,29)(23,30)(25,31)(26,32)(27,28)(33,35,40,48)(34,41,44,45)(36,43,38,46)(37,47,42,39);(17,20)(18,25)(19,27)(21,30)(22,24)(23,28)(26,29)(31,32)(33,36,40,38)(34,42,44,37)(35,46,48,43)(39,41,47,45);(1,3)(2,8)(4,15)(5,16)(6,14)(7,11)(9,13)(10,12)(17,20)(18,25)(19,21)(22,32)(23,28)(24,31)(26,29)(27,30)(33,37)(34,36)(35,47)(38,44)(39,48)(40,42)(41,43)(45,46);(1,4,6,10)(2,9,7,5)(3,12,14,15)(8,16,11,13)(17,21)(18,22)(19,28)(20,30)(23,27)(24,25)(26,31)(29,32)(33,38,40,36)(34,42,44,37)(35,46,48,43)(39,45,47,41);(1,5,6,9)(2,10,7,4)(3,13,14,16)(8,15,11,12)(17,22)(18,21)(19,29)(20,24)(23,31)(25,30)(26,27)(28,32)(33,39,40,47)(34,43,44,46)(35,37,48,42)(36,41,38,45)

Polycyclic presentation:

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

Conjugacy classes & character table

83 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 10319522);