#9206472

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	#2540	C₂²
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

3882 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:

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

Polycyclic presentation:

2^9|a^2=gi,b^2=hi,c^2=gh,d^2=h,e^2=gi,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, 9113668);