#9203729

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

4256 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)(3,7)(4,5)(6,9)(8,10)(11,12)(13,16)(14,15)(17,18,22,20)(19,24,27,23)(21,28,30,25)(26,31,29,32)(33,34)(35,44)(36,40)(37,41)(38,42)(39,43)(45,48)(46,47);(1,3)(2,7)(4,11)(5,12)(6,10)(8,9)(13,14)(15,16)(17,19,22,27)(18,23,20,24)(21,26,30,29)(25,31,28,32)(33,35)(34,38)(36,45)(37,46)(39,42)(40,41)(43,44)(47,48);(3,10)(7,8)(11,13)(12,16)(18,20)(23,24)(25,28)(31,32)(33,36,39,46)(34,40,43,47)(35,45,42,37)(38,41,44,48);(1,3)(2,8)(4,13)(5,12)(6,10)(7,9)(11,14)(15,16)(17,20,22,18)(19,24,27,23)(21,28,30,25)(26,32,29,31)(33,36)(34,40)(35,45)(37,42)(38,41)(39,46)(43,47)(44,48);(1,4,6,14)(2,5,9,15)(3,11,10,13)(7,12,8,16)(17,19)(18,24)(20,23)(21,29)(22,27)(25,31)(26,30)(28,32)(33,37,39,45)(34,41,43,48)(35,46,42,36)(38,40,44,47);(1,5)(2,4)(3,12)(6,15)(7,11)(8,13)(9,14)(10,16)(17,21)(18,25)(19,26)(20,28)(22,30)(23,31)(24,32)(27,29)(33,38,39,44)(34,42,43,35)(36,41,46,48)(37,47,45,40)

Polycyclic presentation:

2^9|a^2=h,b^2=h,c^2=gh,d^2=h,e^2=ghi,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, 9110925);