#8535364

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:

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

Polycyclic presentation:

2^9|a^2=g,b^2=gh,c^2=i,d^2=gh,e^2=hi,f^2=g,b^a=bg,c^a=ch,d^a=dg,e^a=eh,c^b=ci,d^b=dh,e^c=egi,f^d=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, 8442560);