#9206592

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	#2813	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)(4,9)(5,15)(6,8)(7,13)(11,16)(12,14)(17,18,23,21)(19,26,24,20)(22,29,30,25)(27,28,31,32)(35,41)(38,42)(43,47)(45,48);(1,3,6,13)(2,7,8,10)(4,11,14,15)(5,12,16,9)(18,21)(19,24)(25,29)(27,31)(33,34,39,36)(35,38,41,42)(37,44,46,40)(43,48,47,45);(2,8)(4,14)(7,10)(11,15)(17,19)(18,20)(21,26)(22,27)(23,24)(25,28)(29,32)(30,31)(33,35)(34,38)(36,42)(37,43)(39,41)(40,45)(44,48)(46,47);(1,4,6,14)(2,9,8,12)(3,11,13,15)(5,10,16,7)(17,20,23,26)(18,24,21,19)(22,28,30,32)(25,31,29,27)(33,36,39,34)(35,42,41,38)(37,44,46,40)(43,48,47,45);(1,3)(2,10)(4,15)(5,9)(6,13)(7,8)(11,14)(12,16)(17,21,23,18)(19,26,24,20)(22,29,30,25)(27,32,31,28)(33,37,39,46)(34,40,36,44)(35,43,41,47)(38,45,42,48);(1,5)(2,11)(3,12)(4,10)(6,16)(7,14)(8,15)(9,13)(17,22)(18,25)(19,27)(20,28)(21,29)(23,30)(24,31)(26,32)(33,38,39,42)(34,41,36,35)(37,45,46,48)(40,47,44,43)

Polycyclic presentation:

2^9|a^2=i,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, 9113788);