#8095790

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

4111 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:

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

Polycyclic presentation:

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

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, 8002986);