#8888156

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

4052 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:

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

Polycyclic presentation:

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

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, 8795352);