#8095494

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

4053 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:

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

Polycyclic presentation:

2^9|b^2=h,c^2=g,d^2=ghi,e^2=gi,f^2=g,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, 8002690);