#9945507

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

3972 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:

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

Polycyclic presentation:

2^9|a^2=gh,b^2=h,d^2=hi,e^2=gh,f^2=g,b^a=bg,c^a=ch,d^a=di,f^a=fh,c^b=ci,d^b=dgi,e^b=eghi,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, 9852703);