#9717591

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	#2682	C₂²
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

3865 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)(3,6)(4,7)(5,8)(9,10)(11,16)(12,15)(13,14)(17,18)(19,27)(20,21)(22,25)(23,26)(24,30)(28,32)(29,31)(36,44)(40,47)(42,48)(45,46);(1,3)(2,6)(4,10)(5,13)(7,9)(8,14)(11,15)(12,16)(17,19)(18,24)(20,28)(21,29)(22,31)(23,30)(25,32)(26,27)(33,34)(35,37)(36,40)(38,41)(39,43)(42,45)(44,47)(46,48);(2,4)(3,9)(8,12)(11,13)(17,20,23,22)(18,25,26,21)(19,28,30,31)(24,32,27,29)(33,35)(34,39)(36,42)(37,41)(38,43)(40,46)(44,48)(45,47);(1,4,7,2)(3,10,9,6)(5,12,15,8)(11,14,13,16)(17,21,23,25)(18,20,26,22)(19,29,30,32)(24,28,27,31)(33,36)(34,40)(35,42)(37,45)(38,44)(39,46)(41,47)(43,48);(1,5,7,15)(2,8,4,12)(3,11,9,13)(6,16,10,14)(17,22,23,20)(18,25,26,21)(19,28,30,31)(24,29,27,32)(33,37,38,39)(34,35,41,43)(36,45,44,46)(40,42,47,48);(1,6,7,10)(2,9,4,3)(5,14,15,16)(8,11,12,13)(17,19)(18,24)(20,28)(21,29)(22,31)(23,30)(25,32)(26,27)(33,35)(34,37)(36,42)(38,43)(39,41)(40,45)(44,48)(46,47)

Polycyclic presentation:

2^9|c^2=g,d^2=h,e^2=gi,f^2=gh,b^a=bg,c^a=ch,d^a=di,f^a=fh,c^b=ci,e^b=eh,d^c=dh,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, 9624787);