#9210339

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
prime radical	C₂²×D₄	C₂⁴
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

3801 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)(4,11)(5,8)(6,9)(10,15)(12,16)(13,14)(17,18,22,25)(19,26,23,20)(21,24,31,32)(27,30,29,28)(33,34,39,37)(35,40,45,46)(36,42,38,41)(43,48,44,47);(1,3,6,4)(2,7,9,11)(5,10,14,12)(8,15,13,16)(17,19)(18,20)(21,29)(22,23)(24,30)(25,26)(27,31)(28,32)(33,35)(34,40)(36,43)(37,46)(38,44)(39,45)(41,47)(42,48);(3,4)(7,11)(10,12)(15,16)(17,20,22,26)(18,23,25,19)(21,28,31,30)(24,29,32,27)(33,36,39,38)(34,41,37,42)(35,43,45,44)(40,47,46,48);(1,4,6,3)(2,7,9,11)(5,12,14,10)(8,15,13,16)(17,20)(18,19)(21,30)(22,26)(23,25)(24,29)(27,32)(28,31);(1,2)(3,7)(4,11)(5,13)(6,9)(8,14)(10,16)(12,15)(17,21)(18,24)(19,27)(20,28)(22,31)(23,29)(25,32)(26,30)(33,37,39,34)(35,40,45,46)(36,42,38,41)(43,47,44,48);(1,5,6,14)(2,8,9,13)(3,10,4,12)(7,15,11,16)(17,19)(18,20)(21,29)(22,23)(24,30)(25,26)(27,31)(28,32)(33,38,39,36)(34,42,37,41)(35,44,45,43)(40,48,46,47)

Polycyclic presentation:

2^9|a^2=h,b^2=i,c^2=h,d^2=i,e^2=gh,f^2=ghi,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, 9117535);