#9711960

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

3961 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:

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

Polycyclic presentation:

2^9|a^2=hi,c^2=ghi,d^2=g,e^2=gh,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, 9619156);