#9717574

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
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4150 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 48
transitive degree	μ_t(G) = 128

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

Polycyclic presentation:

2^9|c^2=g,d^2=h,e^2=gh,f^2=h,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, 9624770);