#9923885

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
prime radical	#49197	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

3929 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:

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

Polycyclic presentation:

2^9|a^2=g,b^2=gh,c^2=gi,d^2=i,e^2=h,f^2=h,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, 9831081);