#8749757

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	#36677	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

3969 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)(3,5)(4,9)(6,10)(7,11)(8,14)(12,15)(13,16)(17,18,22,25)(19,26,21,23)(20,24,29,32)(27,31,28,30)(33,34,37,38)(35,41,42,39)(36,44,45,40)(43,46,48,47);(1,3)(2,8)(4,12)(5,11)(6,13)(7,14)(9,10)(15,16)(17,19,22,21)(18,23,25,26)(20,28,29,27)(24,31,32,30)(34,38)(35,42)(40,44)(43,48);(1,4,7,13)(2,9,11,16)(3,12,14,6)(5,15,8,10)(17,19,22,21)(18,23,25,26)(20,28,29,27)(24,31,32,30)(33,34)(35,39)(36,40)(37,38)(41,42)(43,46)(44,45)(47,48);(1,4)(2,9)(3,12)(5,15)(6,14)(7,13)(8,10)(11,16)(17,20,22,29)(18,24,25,32)(19,27,21,28)(23,30,26,31)(33,35,37,42)(34,39,38,41)(36,43,45,48)(40,46,44,47);(1,5,7,8)(2,3,11,14)(4,15,13,10)(6,9,12,16)(17,18)(19,23)(20,24)(21,26)(22,25)(27,30)(28,31)(29,32)(33,36)(34,40)(35,43)(37,45)(38,44)(39,46)(41,47)(42,48);(1,6,7,12)(2,10,11,15)(3,13,14,4)(5,16,8,9)(17,21,22,19)(18,23,25,26)(20,27,29,28)(24,31,32,30)

Polycyclic presentation:

2^9|a^2=h,b^2=hi,c^2=gi,d^2=h,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, 8656953);