#8750007

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	#2541	C₂²
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4031 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)(4,7)(5,13)(6,10)(9,15)(11,12)(14,16)(17,18)(19,29)(20,26)(21,22)(23,28)(24,31)(25,32)(27,30)(33,34,39,43)(35,40,38,42)(36,41,47,37)(44,46,45,48);(1,3,7,11)(2,8,4,12)(5,10,15,14)(6,9,16,13)(17,19)(18,24)(20,21)(22,25)(23,31)(26,27)(28,29)(30,32)(33,35,39,38)(34,40,43,42)(36,45,47,44)(37,46,41,48);(2,4)(3,11)(9,13)(10,14)(17,20,23,32)(18,25,28,26)(19,21,31,30)(22,29,27,24)(33,36,39,47)(34,41,43,37)(35,44,38,45)(40,46,42,48);(1,4,7,2)(3,8,11,12)(5,13,15,9)(6,10,16,14)(17,20)(18,26)(19,30)(21,31)(22,24)(23,32)(25,28)(27,29)(33,36)(34,37)(35,45)(38,44)(39,47)(40,46)(41,43)(42,48);(1,5)(2,9)(3,10)(4,13)(6,12)(7,15)(8,16)(11,14)(17,21,23,30)(18,27,28,22)(19,20,31,32)(24,26,29,25)(33,37)(34,36)(35,46)(38,48)(39,41)(40,45)(42,44)(43,47);(1,6,7,16)(2,10,4,14)(3,9,11,13)(5,8,15,12)(17,22,23,27)(18,21,28,30)(19,25,31,26)(20,29,32,24)(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=hi,d^2=gh,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, 8657203);