#9713723

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

4112 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)(3,12)(4,10)(5,11)(6,8)(7,9)(13,16)(14,15)(17,18)(19,24)(20,23)(21,25)(22,30)(26,32)(27,28)(29,31)(34,36)(35,38)(41,44)(43,45);(1,3,7,6)(2,8,9,12)(4,13,14,11)(5,15,16,10)(17,19)(18,24)(20,28)(21,30)(22,25)(23,27)(26,31)(29,32)(33,34,39,36)(35,42,38,40)(37,44,46,41)(43,47,45,48);(2,9)(4,14)(8,12)(11,13)(18,20)(19,27)(25,29)(26,30)(33,35,39,38)(34,40,36,42)(37,43,46,45)(41,47,44,48);(1,4,7,14)(2,10,9,15)(3,13,6,11)(5,12,16,8)(17,20,23,18)(19,28,27,24)(21,29,31,25)(22,30,32,26)(33,36,39,34)(35,40,38,42)(37,44,46,41)(43,47,45,48);(1,5)(2,11)(3,10)(4,12)(6,15)(7,16)(8,14)(9,13)(17,21,23,31)(18,25,20,29)(19,26,27,30)(22,24,32,28)(33,37)(34,41)(35,43)(36,44)(38,45)(39,46)(40,47)(42,48);(1,6,7,3)(2,12,9,8)(4,11,14,13)(5,10,16,15)(17,22)(18,26)(19,25)(20,30)(21,28)(23,32)(24,31)(27,29)(33,38,39,35)(34,40,36,42)(37,45,46,43)(41,47,44,48)

Polycyclic presentation:

2^9|b^2=g,c^2=ghi,d^2=h,e^2=gh,f^2=g,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, 9620919);