#9724839

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	#50792	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4003 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)(4,10)(5,11)(6,16)(7,9)(12,13)(14,15)(17,18)(19,20)(21,29)(22,26)(23,24)(25,31)(27,32)(28,30)(33,34)(35,41)(36,46)(37,40)(38,42)(39,45)(43,47)(44,48);(1,3,7,6)(2,8,9,16)(4,12,11,14)(5,15,10,13)(17,19,22,24)(18,23,26,20)(21,27,31,30)(25,32,29,28)(33,35,40,45)(34,41,37,39)(36,44,38,43)(42,47,46,48);(2,9)(4,11)(8,16)(12,14)(17,20,22,23)(18,24,26,19)(21,28,31,32)(25,30,29,27)(33,36,40,38)(34,42,37,46)(35,43,45,44)(39,47,41,48);(1,4,7,11)(2,10,9,5)(3,12,6,14)(8,13,16,15)(17,20)(18,19)(21,28)(22,23)(24,26)(25,27)(29,30)(31,32)(33,37,40,34)(35,39,45,41)(36,42,38,46)(43,48,44,47);(1,5,7,10)(2,11,9,4)(3,13,6,15)(8,12,16,14)(17,19)(18,20)(21,30)(22,24)(23,26)(25,32)(27,31)(28,29)(33,38,40,36)(34,42,37,46)(35,44,45,43)(39,47,41,48);(1,6,7,3)(2,8,9,16)(4,14,11,12)(5,15,10,13)(17,21)(18,25)(19,27)(20,28)(22,31)(23,32)(24,30)(26,29)(33,39)(34,35)(36,47)(37,45)(38,48)(40,41)(42,43)(44,46)

Polycyclic presentation:

2^9|b^2=h,c^2=gi,d^2=gh,e^2=gh,f^2=ghi,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, 9632035);