#8188231

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	#56800	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4037 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)(3,9,4,8)(5,13,14,10)(11,15,12,16)(17,18)(19,26)(20,23)(21,24)(22,25)(27,32)(28,30)(29,31)(33,34,38,39)(35,37,41,46)(36,43,45,40)(42,48,47,44);(1,3,7,4)(2,8,6,9)(5,11,14,12)(10,15,13,16)(17,19,23,21)(18,24,20,26)(22,27,30,29)(25,31,28,32)(33,34,38,39)(35,37,41,46)(36,40,45,43)(42,44,47,48);(2,6)(8,9)(10,13)(15,16)(22,30)(25,28)(27,29)(31,32)(34,39)(37,46)(40,43)(44,48);(1,4,7,3)(2,9,6,8)(5,12,14,11)(10,16,13,15)(17,20,23,18)(19,24,21,26)(22,28,30,25)(27,31,29,32)(33,35,38,41)(34,37,39,46)(36,42,45,47)(40,44,43,48);(1,5)(2,10)(3,11)(4,12)(6,13)(7,14)(8,15)(9,16)(17,21,23,19)(18,24,20,26)(22,29,30,27)(25,31,28,32)(33,36,38,45)(34,40,39,43)(35,42,41,47)(37,44,46,48);(1,6,7,2)(3,9,4,8)(5,13,14,10)(11,16,12,15)(17,22,23,30)(18,25,20,28)(19,27,21,29)(24,31,26,32)(33,37)(34,41)(35,39)(36,44)(38,46)(40,47)(42,43)(45,48)

Polycyclic presentation:

2^9|a^2=h,b^2=gi,d^2=gi,e^2=gh,f^2=g,b^a=bg,c^a=ch,d^a=dg,e^a=eh,c^b=ci,d^b=dghi,f^c=fgi

Conjugacy classes & character table

92 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 8095427);