#9173141

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

4265 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:

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

Polycyclic presentation:

2^9|a^2=hi,b^2=h,c^2=h,d^2=h,f^2=gh,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, 9080337);