#8724828

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

4038 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)(5,10)(6,8)(9,14)(11,15)(13,16)(33,34)(35,44)(36,46)(37,45)(38,41)(39,40)(42,47)(43,48);(1,3,6,10)(2,7,8,5)(4,11,14,16)(9,13,12,15)(17,18)(19,24)(20,25)(21,26)(22,23)(27,31)(28,32)(29,30)(34,40)(36,47)(41,44)(43,45);(2,8)(3,10)(9,12)(11,16)(18,23)(19,27)(25,29)(26,28)(33,35,39,38)(34,41,40,44)(36,43,47,45)(37,42,48,46);(1,3)(2,7)(4,11)(5,8)(6,10)(9,13)(12,15)(14,16)(17,19,22,27)(18,24,23,31)(20,28,30,26)(21,25,32,29)(33,36,39,47)(34,42,40,46)(35,45,38,43)(37,44,48,41);(1,4,6,14)(2,9,8,12)(3,11,10,16)(5,15,7,13)(17,20)(18,25)(19,28)(21,31)(22,30)(23,29)(24,32)(26,27)(33,37)(34,43)(35,42)(36,44)(38,46)(39,48)(40,45)(41,47);(1,5,6,7)(2,10,8,3)(4,13,14,15)(9,11,12,16)(17,21,22,32)(18,26,23,28)(19,29,27,25)(20,24,30,31)(33,38,39,35)(34,41,40,44)(36,45,47,43)(37,42,48,46)

Polycyclic presentation:

2^9|b^2=gh,c^2=g,d^2=hi,e^2=gh,f^2=gi,b^a=bg,c^a=ch,d^a=dg,e^a=eh,c^b=ci,d^b=dh,d^c=dgi,f^d=fgi,f^e=fgi

Conjugacy classes & character table

89 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 8632024);