#9206646

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	#2699	C₂²
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

3799 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)(3,10)(4,7)(5,13)(6,9)(8,14)(11,12)(15,16)(19,22)(24,26)(27,29)(31,32)(33,34,39,37)(35,38,42,40)(36,44,46,41)(43,47,48,45);(1,2,6,9)(3,4,7,10)(5,8,14,13)(11,12,15,16)(17,18,23,20)(19,24,22,26)(21,28,30,25)(27,32,29,31)(34,37)(35,42)(41,44)(43,48);(1,3,6,7)(2,4,9,10)(5,11,14,15)(8,12,13,16)(17,19,23,22)(18,24,20,26)(21,27,30,29)(25,31,28,32)(33,35,39,42)(34,40,37,38)(36,43,46,48)(41,47,44,45);(1,4,6,10)(2,7,9,3)(5,12,14,16)(8,15,13,11)(17,20,23,18)(19,26,22,24)(21,28,30,25)(27,32,29,31)(33,36,39,46)(34,41,37,44)(35,43,42,48)(38,45,40,47);(1,2)(3,4)(5,13)(6,9)(7,10)(8,14)(11,16)(12,15)(17,21,23,30)(18,25,20,28)(19,27,22,29)(24,31,26,32)(33,37,39,34)(35,38,42,40)(36,44,46,41)(43,45,48,47);(1,5)(2,8)(3,11)(4,12)(6,14)(7,15)(9,13)(10,16)(17,22,23,19)(18,26,20,24)(21,29,30,27)(25,32,28,31)(33,38)(34,35)(36,45)(37,42)(39,40)(41,43)(44,48)(46,47)

Polycyclic presentation:

2^9|a^2=i,b^2=hi,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, 9113842);