#8749886

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	#2414	C₂²
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

3941 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)(4,12)(5,10)(6,7)(9,14)(11,13)(15,16)(17,18)(19,23)(20,26)(21,25)(22,24)(27,30)(28,29)(31,32)(33,34,39,43)(35,40,38,42)(36,41,47,37)(44,46,45,48);(2,7)(3,11)(9,12)(10,16)(17,19,22,26)(18,23,24,20)(21,27,31,29)(25,30,32,28)(33,35,39,38)(34,40,43,42)(36,45,47,44)(37,46,41,48);(2,7)(3,11)(9,12)(10,16)(18,24)(19,26)(25,32)(27,29)(33,36,39,47)(34,41,43,37)(35,44,38,45)(40,46,42,48);(1,3,6,11)(2,8,7,13)(4,10,14,16)(5,9,15,12)(17,19)(18,20)(21,29)(22,26)(23,24)(25,30)(27,31)(28,32)(33,36)(34,37)(35,45)(38,44)(39,47)(40,46)(41,43)(42,48);(1,4,6,14)(2,9,7,12)(3,10,11,16)(5,8,15,13)(17,20)(18,19)(21,30)(22,23)(24,26)(25,29)(27,32)(28,31)(33,37)(34,36)(35,46)(38,48)(39,41)(40,45)(42,44)(43,47);(1,5,6,15)(2,10,7,16)(3,12,11,9)(4,13,14,8)(17,21,22,31)(18,25,24,32)(19,27,26,29)(20,28,23,30)(33,38,39,35)(34,42,43,40)(36,45,47,44)(37,46,41,48)

Polycyclic presentation:

2^9|a^2=ghi,b^2=i,c^2=ghi,d^2=g,e^2=g,f^2=gi,b^a=bg,c^a=ch,d^a=di,e^a=eh,c^b=ci,d^b=dh,d^c=dgi,f^d=fgi,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, 8657082);