#9720447

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	#2703	C₂²
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

3880 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)(3,11)(4,9)(5,15)(6,8)(7,12)(10,13)(14,16)(18,23)(19,26)(21,29)(28,31)(33,34,38,40)(35,36,42,44)(37,39,47,46)(41,43,45,48);(1,3,6,12)(2,7,8,11)(4,13,14,5)(9,15,16,10)(17,18,22,23)(19,20,26,24)(21,25,29,30)(27,28,32,31)(33,35)(34,36)(37,45)(38,42)(39,48)(40,44)(41,47)(43,46);(2,8)(7,11)(9,16)(10,15)(17,19,22,26)(18,24,23,20)(21,27,29,32)(25,31,30,28)(33,36,38,44)(34,35,40,42)(37,43,47,48)(39,41,46,45);(1,2)(3,7)(4,9)(5,10)(6,8)(11,12)(13,15)(14,16)(17,19)(18,20)(21,27)(22,26)(23,24)(25,28)(29,32)(30,31)(33,36)(34,35)(37,43)(38,44)(39,41)(40,42)(45,46)(47,48);(1,4,6,14)(2,9,8,16)(3,5,12,13)(7,10,11,15)(17,20)(18,19)(21,28)(22,24)(23,26)(25,27)(29,31)(30,32)(33,37)(34,39)(35,41)(36,43)(38,47)(40,46)(42,45)(44,48);(1,5)(2,10)(3,4)(6,13)(7,9)(8,15)(11,16)(12,14)(17,21,22,29)(18,25,23,30)(19,27,26,32)(20,28,24,31)(33,34)(35,36)(37,46)(38,40)(39,47)(41,48)(42,44)(43,45)

Polycyclic presentation:

2^9|a^2=gh,b^2=g,c^2=i,e^2=hi,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, 9627643);