#8745752

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
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4119 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 48
transitive degree	μ_t(G) = 128

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

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

Polycyclic presentation:

2^9|a^2=gi,b^2=hi,c^2=hi,d^2=i,f^2=gh,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, 8652948);