#8097115

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

4114 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)(3,10)(4,8)(5,7)(6,9)(11,16)(12,15)(13,14)(18,20)(24,25)(26,28)(31,32)(33,34,39,44)(35,37,47,40)(36,42,46,41)(38,43,45,48);(4,12)(8,15)(11,13)(14,16)(17,18,23,20)(19,24,21,25)(22,26,30,28)(27,31,29,32)(33,35,39,47)(34,40,44,37)(36,38,46,45)(41,43,42,48);(1,3,6,5)(2,7,9,10)(4,11,12,13)(8,14,15,16)(17,19,23,21)(18,24,20,25)(22,27,30,29)(26,31,28,32)(33,36)(34,41)(35,45)(37,43)(38,47)(39,46)(40,48)(42,44);(1,4)(2,8)(3,11)(5,13)(6,12)(7,14)(9,15)(10,16)(17,20,23,18)(19,25,21,24)(22,28,30,26)(27,32,29,31)(33,37,39,40)(34,35,44,47)(36,43,46,48)(38,41,45,42);(1,5,6,3)(2,7,9,10)(4,13,12,11)(8,14,15,16)(17,21,23,19)(18,25,20,24)(22,27,30,29)(26,31,28,32)(33,36)(34,42)(35,38)(37,43)(39,46)(40,48)(41,44)(45,47);(1,2)(3,7)(4,8)(5,10)(6,9)(11,14)(12,15)(13,16)(17,22)(18,26)(19,27)(20,28)(21,29)(23,30)(24,31)(25,32)(33,38)(34,43)(35,46)(36,47)(37,41)(39,45)(40,42)(44,48)

Polycyclic presentation:

2^9|a^2=i,b^2=g,c^2=gh,d^2=g,e^2=gh,b^a=bg,c^a=ch,d^a=dg,c^b=ci,d^b=dh,f^e=fghi

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, 8004311);