#9912884

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

3889 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)(3,8)(4,7)(5,9)(6,10)(11,16)(12,15)(13,14)(17,18,21,24)(19,22,25,29)(20,23,28,32)(26,30,27,31)(33,34,39,36)(35,42,37,40)(38,44,46,41)(43,47,45,48);(1,3,6,4)(2,7,10,8)(5,11,14,12)(9,15,13,16)(19,25)(22,29)(26,27)(30,31)(33,34,39,36)(35,40,37,42)(38,41,46,44)(43,47,45,48);(1,3,6,4)(2,8,10,7)(5,11,14,12)(9,16,13,15)(17,19)(18,22)(20,26)(21,25)(23,30)(24,29)(27,28)(31,32)(33,35,39,37)(34,40,36,42)(38,43,46,45)(41,47,44,48);(1,4,6,3)(2,7,10,8)(5,12,14,11)(9,15,13,16)(17,18)(19,22)(20,23)(21,24)(25,29)(26,30)(27,31)(28,32)(33,36,39,34)(35,40,37,42)(38,44,46,41)(43,47,45,48);(1,2)(3,8)(4,7)(5,13)(6,10)(9,14)(11,15)(12,16)(17,19)(18,22)(20,27)(21,25)(23,31)(24,29)(26,28)(30,32)(33,37,39,35)(34,40,36,42)(38,45,46,43)(41,47,44,48);(1,5,6,14)(2,9,10,13)(3,11,4,12)(7,15,8,16)(17,20)(18,23)(19,26)(21,28)(22,30)(24,32)(25,27)(29,31)(33,38)(34,41)(35,43)(36,44)(37,45)(39,46)(40,47)(42,48)

Polycyclic presentation:

2^9|a^2=hi,b^2=gh,c^2=gh,d^2=gh,e^2=h,f^2=g,b^a=bg,c^a=ch,d^a=di,f^a=fh,c^b=ci,e^b=eghi,d^c=dh,f^e=fgi

Conjugacy classes & character table

77 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 9820080);