#8750246

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, prime radical, cosocle	C₂³	C₂⁶
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

3743 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 48
transitive degree	μ_t(G) = 512

A minimal faithful permutation representation — 6 generators on 48 points:

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

Polycyclic presentation:

2^9|a^2=hi,b^2=i,c^2=ghi,d^2=gh,e^2=h,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, 8657442);