#8095296

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
prime radical	C₂⁴	C₂⁵
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

3920 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)(3,7,9,10)(5,12,13,8)(11,16,15,14)(18,19)(21,29)(23,25)(28,32)(33,34,39,40)(35,38,36,41)(37,42,46,47)(43,45,44,48);(3,9)(7,10)(11,15)(14,16)(17,18,22,19)(20,23,27,25)(21,24,29,26)(28,30,32,31)(34,40)(35,36)(42,47)(43,44);(2,4)(7,10)(8,12)(14,16)(20,27)(23,25)(28,32)(30,31)(33,35,39,36)(34,41,40,38)(37,44,46,43)(42,45,47,48);(1,3)(2,7)(4,10)(5,11)(6,9)(8,14)(12,16)(13,15)(17,19,22,18)(20,25,27,23)(21,26,29,24)(28,31,32,30)(33,36,39,35)(34,38,40,41)(37,44,46,43)(42,45,47,48);(1,4,6,2)(3,10,9,7)(5,12,13,8)(11,16,15,14)(17,20,22,27)(18,23,19,25)(21,28,29,32)(24,30,26,31)(33,37,39,46)(34,42,40,47)(35,43,36,44)(38,45,41,48);(1,5)(2,8)(3,11)(4,12)(6,13)(7,14)(9,15)(10,16)(17,21)(18,24)(19,26)(20,28)(22,29)(23,30)(25,31)(27,32)(33,38)(34,35)(36,40)(37,45)(39,41)(42,43)(44,47)(46,48)

Polycyclic presentation:

2^9|a^2=h,b^2=gi,c^2=i,d^2=g,e^2=ghi,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, 8002492);