#9206593

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	#2635	C₂²
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

3803 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:

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

Polycyclic presentation:

2^9|a^2=gi,b^2=hi,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, 9113789);