#9726611

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

4091 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)(3,5)(4,11)(6,14)(7,8)(9,16)(10,15)(12,13)(17,18)(19,27)(20,25)(21,31)(22,24)(23,28)(26,29)(30,32)(33,34)(35,45)(36,39)(37,42)(38,43)(40,44)(41,48)(46,47);(1,3,7,4)(2,5,8,11)(6,10,16,12)(9,13,14,15)(17,19,22,28)(18,23,24,27)(20,29,30,21)(25,31,32,26)(33,35)(34,41)(36,38)(37,46)(39,47)(40,48)(42,43)(44,45);(2,8)(3,4)(9,14)(10,12)(18,24)(23,27)(25,32)(26,31)(33,36,40,37)(34,39,44,42)(35,38,48,46)(41,47,45,43);(1,4,7,3)(2,5,8,11)(6,12,16,10)(9,13,14,15)(17,18)(19,23)(20,25)(21,26)(22,24)(27,28)(29,31)(30,32)(33,37,40,36)(34,42,44,39)(35,46,48,38)(41,43,45,47);(1,5)(2,3)(4,8)(6,15)(7,11)(9,12)(10,14)(13,16)(17,20,22,30)(18,25,24,32)(19,21,28,29)(23,26,27,31)(33,38,40,46)(34,43,44,47)(35,36,48,37)(39,41,42,45);(1,6,7,16)(2,9,8,14)(3,10,4,12)(5,13,11,15)(17,21)(18,26)(19,20)(22,29)(23,25)(24,31)(27,32)(28,30)(33,39,40,42)(34,36,44,37)(35,47,48,43)(38,45,46,41)

Polycyclic presentation:

2^9|b^2=h,c^2=ghi,d^2=gh,e^2=g,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, 9633807);