#8750279

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	#2813	C₂²
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

3883 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:

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

Polycyclic presentation:

2^9|a^2=gi,b^2=hi,c^2=ghi,d^2=gh,e^2=gh,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, 8657475);