#8459471

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

4113 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:

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

Polycyclic presentation:

2^9|b^2=gh,c^2=gh,e^2=i,f^2=hi,b^a=bg,c^a=ch,d^a=dg,e^a=eghi,c^b=ci,d^b=dh,f^d=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, 8366667);