#8508987

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

4166 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)(3,9)(4,8)(5,16)(6,10)(7,11)(12,15)(13,14)(17,18)(19,23)(20,29)(21,24)(22,25)(26,32)(27,30)(28,31)(37,45)(41,46)(43,44)(47,48);(1,3,7,4)(2,8,11,9)(5,14,10,12)(6,13,16,15)(17,19,22,28)(18,23,25,31)(20,27,21,26)(24,32,29,30)(33,34,39,38)(35,42,36,40)(37,46,45,41)(43,47,44,48);(1,3,7,4)(2,9,11,8)(5,12,10,14)(6,13,16,15)(17,20,22,21)(18,24,25,29)(19,26,28,27)(23,30,31,32)(33,35,39,36)(34,40,38,42)(37,43,45,44)(41,47,46,48);(1,4,7,3)(2,9,11,8)(5,14,10,12)(6,13,16,15)(17,21,22,20)(18,24,25,29)(19,27,28,26)(23,30,31,32)(33,36,39,35)(34,40,38,42)(37,44,45,43)(41,47,46,48);(1,5,7,10)(2,6,11,16)(3,12,4,14)(8,15,9,13)(17,19)(18,23)(20,26)(21,27)(22,28)(24,30)(25,31)(29,32)(33,37,39,45)(34,41,38,46)(35,43,36,44)(40,47,42,48);(1,6)(2,10)(3,13)(4,15)(5,11)(7,16)(8,12)(9,14)(17,18)(19,23)(20,24)(21,29)(22,25)(26,30)(27,32)(28,31)(33,38,39,34)(35,42,36,40)(37,46,45,41)(43,48,44,47)

Polycyclic presentation:

2^9|b^2=gi,c^2=gi,d^2=gi,e^2=ghi,f^2=hi,b^a=bg,c^a=ch,d^a=dg,e^a=eghi,c^b=ci,d^b=dh,e^b=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, 8416183);