#9712170

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	#2828	C₂²
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

3863 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)(3,4)(5,13)(6,7)(8,10)(9,15)(11,14)(12,16)(17,18,23,20)(19,24,26,21)(22,28,31,25)(27,29,32,30)(33,34,39,37)(35,41,36,40)(38,42,46,45)(43,48,44,47);(2,7)(4,8)(9,13)(12,14)(17,19)(18,24)(20,21)(22,27)(23,26)(25,30)(28,29)(31,32)(33,35,39,36)(34,40,37,41)(38,43,46,44)(42,47,45,48);(2,7)(3,10)(9,13)(11,16)(18,20)(19,26)(25,28)(27,32)(33,34,39,37)(35,40,36,41)(38,42,46,45)(43,47,44,48);(1,3)(2,8)(4,7)(5,11)(6,10)(9,14)(12,13)(15,16)(17,20,23,18)(19,21,26,24)(22,28,31,25)(27,29,32,30)(33,36,39,35)(34,41,37,40)(38,44,46,43)(42,48,45,47);(1,4,6,8)(2,3,7,10)(5,14,15,12)(9,16,13,11)(17,21)(18,19)(20,26)(22,30)(23,24)(25,32)(27,28)(29,31)(33,37,39,34)(35,41,36,40)(38,42,46,45)(43,47,44,48);(1,5,6,15)(2,9,7,13)(3,11,10,16)(4,12,8,14)(17,22)(18,25)(19,27)(20,28)(21,29)(23,31)(24,30)(26,32)(33,38)(34,42)(35,43)(36,44)(37,45)(39,46)(40,47)(41,48)

Polycyclic presentation:

2^9|a^2=gh,b^2=ghi,c^2=ghi,d^2=gh,e^2=g,f^2=hi,b^a=bg,c^a=ch,d^a=dg,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, 9619366);