#9717579

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

4097 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)(3,8)(4,7)(5,9)(6,14)(10,16)(11,12)(13,15)(20,28)(23,30)(27,32)(29,31)(34,40)(35,38)(36,42)(45,47);(1,3)(2,8)(4,12)(5,14)(6,9)(7,11)(10,15)(13,16)(17,18)(19,24)(20,23)(21,25)(22,26)(27,31)(28,30)(29,32)(33,34)(35,43)(36,37)(38,41)(39,40)(42,48)(44,45)(46,47);(2,4)(3,11)(9,13)(10,14)(17,19)(18,21)(20,27)(22,25)(23,29)(24,26)(28,32)(30,31)(33,35,39,38)(34,41,40,43)(36,46,42,44)(37,45,48,47);(1,4,7,2)(3,12,11,8)(5,13,15,9)(6,14,16,10)(17,20)(18,23)(19,27)(21,29)(22,28)(24,31)(25,32)(26,30)(33,36,39,42)(34,37,40,48)(35,44,38,46)(41,47,43,45);(1,5,7,15)(2,9,4,13)(3,10,11,14)(6,8,16,12)(17,19)(18,24)(20,27)(21,26)(22,25)(23,31)(28,32)(29,30)(33,37)(34,42)(35,45)(36,40)(38,47)(39,48)(41,44)(43,46);(1,6)(2,10)(3,9)(4,14)(5,12)(7,16)(8,15)(11,13)(17,21,22,24)(18,25,26,19)(20,29,28,31)(23,32,30,27)(33,38,39,35)(34,41,40,43)(36,46,42,44)(37,47,48,45)

Polycyclic presentation:

2^9|c^2=g,d^2=h,e^2=gh,f^2=ghi,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, 9624775);