#9203632

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

4111 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 48
transitive degree	μ_t(G) = 256

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

Polycyclic presentation:

2^9|c^2=h,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, 9110828);