#8750741

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

3947 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:

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

Polycyclic presentation:

2^9|a^2=gi,b^2=gh,c^2=h,d^2=gi,e^2=g,f^2=gi,b^a=bg,c^a=ch,d^a=di,e^a=eh,c^b=ci,d^b=dh,d^c=dgi,f^d=fgi,f^e=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, 8657937);