#9206645

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	#2676	C₂²
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

3878 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:

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

Polycyclic presentation:

2^9|a^2=gi,b^2=ghi,c^2=gh,d^2=gh,e^2=ghi,f^2=h,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, 9113841);