#9175016

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

4283 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)(3,8,11,15)(4,9,14,13)(5,12,16,10)(17,18)(19,23)(20,26)(21,24)(22,25)(27,31)(28,32)(29,30)(33,34,39,38)(35,43,45,40)(36,41,44,46)(37,42,47,48);(2,7)(4,14)(8,15)(10,12)(17,19,22,26)(18,23,25,20)(21,27,30,32)(24,31,29,28)(33,35,39,45)(34,40,38,43)(36,37,44,47)(41,48,46,42);(2,7)(3,11)(9,13)(10,12)(19,26)(20,23)(27,32)(28,31)(33,36,39,44)(34,41,38,46)(35,37,45,47)(40,48,43,42);(1,3,6,11)(2,8,7,15)(4,10,14,12)(5,13,16,9)(17,20)(18,19)(21,28)(22,23)(24,27)(25,26)(29,32)(30,31)(33,36)(34,41)(35,37)(38,46)(39,44)(40,48)(42,43)(45,47);(1,4)(2,9)(3,12)(5,15)(6,14)(7,13)(8,16)(10,11)(17,18)(19,23)(20,26)(21,29)(22,25)(24,30)(27,28)(31,32)(33,37)(34,42)(35,44)(36,45)(38,48)(39,47)(40,41)(43,46);(1,5,6,16)(2,10,7,12)(3,13,11,9)(4,8,14,15)(17,21)(18,24)(19,27)(20,28)(22,30)(23,31)(25,29)(26,32)(33,38,39,34)(35,43,45,40)(36,46,44,41)(37,42,47,48)

Polycyclic presentation:

2^9|a^2=g,b^2=ghi,c^2=gh,d^2=h,f^2=g,b^a=bg,c^a=ch,d^a=di,f^a=fh,c^b=ci,e^b=eh,d^c=dgi,e^d=egi,f^e=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, 9082212);