#9712143

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	#2671	C₂²
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

3867 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)(5,13)(6,7)(8,12)(10,15)(14,16)(17,18,22,23)(19,21,25,31)(20,24,29,32)(26,30,27,28)(33,34,39,37)(35,41,36,40)(38,42,46,45)(43,48,44,47);(2,7)(4,14)(8,11)(10,13)(17,19)(18,21)(20,27)(22,25)(23,31)(24,28)(26,29)(30,32)(33,35,39,36)(34,40,37,41)(38,43,46,44)(42,47,45,48);(2,7)(3,12)(9,16)(10,13)(18,23)(19,25)(24,32)(26,27)(33,34,39,37)(35,40,36,41)(38,42,46,45)(43,47,44,48);(1,3,6,12)(2,8,7,11)(4,13,14,10)(5,9,15,16)(17,19)(18,21)(20,26)(22,25)(23,31)(24,30)(27,29)(28,32)(33,36,39,35)(34,41,37,40)(38,44,46,43)(42,48,45,47);(1,4,6,14)(2,9,7,16)(3,13,12,10)(5,8,15,11)(17,20)(18,24)(19,26)(21,30)(22,29)(23,32)(25,27)(28,31)(33,37,39,34)(35,41,36,40)(38,42,46,45)(43,47,44,48);(1,5,6,15)(2,10,7,13)(3,9,12,16)(4,11,14,8)(17,21)(18,19)(20,28)(22,31)(23,25)(24,27)(26,32)(29,30)(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=g,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, 9619339);