#8459303

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

4084 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)(4,8)(5,9)(6,10)(11,14)(12,15)(13,16)(17,18)(19,26)(20,25)(21,30)(22,24)(23,27)(28,32)(29,31)(36,44)(40,47)(43,48)(45,46);(3,10)(6,7)(11,16)(13,14)(17,19,23,22)(18,24,27,26)(20,28,21,29)(25,31,30,32)(33,34)(35,39)(36,40)(37,41)(38,42)(43,46)(44,47)(45,48);(1,3)(2,6)(4,11)(5,10)(7,9)(8,13)(12,16)(14,15)(17,20,23,21)(18,25,27,30)(19,28,22,29)(24,31,26,32)(33,35)(34,37)(36,43)(38,41)(39,42)(40,45)(44,48)(46,47);(1,3)(2,7)(4,11)(5,10)(6,9)(8,14)(12,16)(13,15)(17,21,23,20)(18,25,27,30)(19,29,22,28)(24,31,26,32)(33,35)(34,39)(36,43)(37,42)(38,41)(40,46)(44,48)(45,47);(1,4,5,12)(2,8,9,15)(3,11,10,16)(6,13,7,14)(17,22,23,19)(18,26,27,24)(20,29,21,28)(25,32,30,31)(33,36,38,44)(34,40,42,47)(35,43,41,48)(37,45,39,46);(1,2)(3,6)(4,8)(5,9)(7,10)(11,13)(12,15)(14,16)(17,18)(19,24)(20,25)(21,30)(22,26)(23,27)(28,31)(29,32)(33,37,38,39)(34,41,42,35)(36,45,44,46)(40,48,47,43)

Polycyclic presentation:

2^9|b^2=g,c^2=g,d^2=g,e^2=gi,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, 8366499);