#8731245

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

4247 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 48
transitive degree	μ_t(G) = 128

A minimal faithful permutation representation — 6 generators on 48 points:

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

Polycyclic presentation:

2^9|a^2=hi,c^2=gh,d^2=gh,e^2=g,b^a=bg,c^a=ch,d^a=di,e^a=eh,c^b=ci,d^b=dh,d^c=dgi,f^d=fgi,f^e=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, 8638441);