#8750631

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	#45020	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4015 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)(3,7)(4,12)(5,9)(6,10)(8,13)(11,14)(15,16)(17,18)(19,25)(20,23)(21,24)(22,27)(26,32)(28,30)(29,31)(33,34,39,43)(35,41,37,40)(36,42,47,38)(44,45,46,48);(1,3)(2,7)(4,9)(5,12)(6,11)(8,16)(10,14)(13,15)(17,18,21,24)(19,22,27,25)(20,23,30,28)(26,31,29,32)(33,35,39,37)(34,40,43,41)(36,44,47,46)(38,45,42,48);(3,11)(4,13)(7,14)(8,12)(17,18,21,24)(19,22,27,25)(20,23,30,28)(26,31,29,32)(33,35,39,37)(34,41,43,40)(36,46,47,44)(38,45,42,48);(1,4,6,13)(2,8,10,12)(3,9,11,15)(5,7,16,14)(17,18)(19,22)(20,28)(21,24)(23,30)(25,27)(26,32)(29,31)(33,36,39,47)(34,42,43,38)(35,44,37,46)(40,48,41,45);(1,2)(3,7)(4,8)(5,15)(6,10)(9,16)(11,14)(12,13)(17,19)(18,22)(20,29)(21,27)(23,32)(24,25)(26,30)(28,31)(33,37,39,35)(34,40,43,41)(36,46,47,44)(38,45,42,48);(1,5,6,16)(2,9,10,15)(3,12,11,8)(4,14,13,7)(17,20,21,30)(18,23,24,28)(19,26,27,29)(22,31,25,32)(33,38,39,42)(34,36,43,47)(35,45,37,48)(40,44,41,46)

Polycyclic presentation:

2^9|a^2=gh,b^2=g,c^2=g,d^2=ghi,e^2=gh,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, 8657827);