#8383791

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

4051 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)(3,5)(4,8)(6,9)(7,10)(11,13)(12,15)(14,16)(17,18,23,28)(19,24,30,31)(20,25,21,26)(22,27,32,29)(33,34,39,37)(35,36,43,40)(38,41,46,45)(42,44,48,47);(3,4)(5,8)(11,12)(13,15)(17,19,23,30)(18,24,28,31)(20,27,21,29)(22,25,32,26)(33,35,39,43)(34,40,37,36)(38,42,46,48)(41,47,45,44);(1,3,7,4)(2,8,10,5)(6,12,14,11)(9,13,16,15)(17,20,23,21)(18,25,28,26)(19,29,30,27)(22,31,32,24)(33,34,39,37)(35,40,43,36)(38,45,46,41)(42,44,48,47);(1,4,7,3)(2,8,10,5)(6,12,14,11)(9,15,16,13)(17,21,23,20)(18,26,28,25)(19,29,30,27)(22,31,32,24)(33,36)(34,35)(37,43)(38,44)(39,40)(41,42)(45,48)(46,47);(1,5,7,8)(2,4,10,3)(6,13,14,15)(9,12,16,11)(17,19)(18,24)(20,29)(21,27)(22,25)(23,30)(26,32)(28,31)(33,37,39,34)(35,40,43,36)(38,45,46,41)(42,47,48,44);(1,6)(2,9)(3,11)(4,12)(5,13)(7,14)(8,15)(10,16)(17,22,23,32)(18,27,28,29)(19,25,30,26)(20,24,21,31)(33,38,39,46)(34,41,37,45)(35,42,43,48)(36,44,40,47)

Polycyclic presentation:

2^9|a^2=ghi,b^2=ghi,c^2=gi,d^2=i,e^2=gh,f^2=ghi,b^a=bg,c^a=ch,d^a=dg,e^a=eh,c^b=ci,d^b=dh,e^b=egi,f^c=fgi,e^d=egi

Conjugacy classes & character table

86 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 8290987);