#8746073

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

4079 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:

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

Polycyclic presentation:

2^9|a^2=gi,b^2=gi,c^2=i,d^2=hi,e^2=gh,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, 8653269);