#8726624

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

4081 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

89 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 8633820);