#9711912

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

3945 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)(3,10)(4,7)(5,9)(6,8)(11,16)(12,14)(13,15)(17,18)(19,22)(20,27)(21,24)(23,28)(25,31)(26,30)(29,32)(33,34,38,37)(35,39,41,43)(36,40,44,45)(42,46,47,48);(1,3,7,6)(2,8,4,10)(5,13,14,11)(9,16,12,15)(17,19,21,26)(18,22,24,30)(20,25,28,32)(23,29,27,31)(35,41)(39,43)(42,47)(46,48);(2,4)(8,10)(9,12)(15,16)(17,18,21,24)(19,22,26,30)(20,23,28,27)(25,29,32,31)(33,35,38,41)(34,39,37,43)(36,42,44,47)(40,46,45,48);(1,4,7,2)(3,10,6,8)(5,12,14,9)(11,16,13,15)(17,18)(19,22)(20,23)(21,24)(25,29)(26,30)(27,28)(31,32);(1,5,7,14)(2,9,4,12)(3,11,6,13)(8,15,10,16)(17,19)(18,22)(20,25)(21,26)(23,29)(24,30)(27,31)(28,32)(33,36)(34,40)(35,42)(37,45)(38,44)(39,46)(41,47)(43,48);(1,6,7,3)(2,10,4,8)(5,11,14,13)(9,15,12,16)(17,20,21,28)(18,23,24,27)(19,25,26,32)(22,29,30,31)(33,37,38,34)(35,43,41,39)(36,40,44,45)(42,46,47,48)

Polycyclic presentation:

2^9|a^2=i,b^2=h,c^2=ghi,d^2=g,e^2=g,f^2=hi,b^a=bg,c^a=ch,d^a=dg,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, 9619108);