#8726384

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	#42629	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4029 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)(3,8,10,4)(6,9,15,13)(11,14,16,12)(18,23)(19,28)(20,27)(30,31)(35,37)(40,42)(43,45)(47,48);(1,3,7,10)(2,8,5,4)(6,11,15,16)(9,14,13,12)(17,18)(19,24)(20,26)(21,27)(22,23)(25,28)(29,31)(30,32)(33,34)(35,42)(36,39)(37,40)(38,41)(43,48)(44,46)(45,47);(3,10)(4,8)(11,16)(12,14)(17,19,22,28)(18,24,23,25)(20,29,27,32)(21,30,26,31)(33,35,39,37)(34,40,36,42)(38,43,46,45)(41,47,44,48);(1,4)(2,3)(5,10)(6,14)(7,8)(9,16)(11,13)(12,15)(17,19)(18,25)(20,29)(21,31)(22,28)(23,24)(26,30)(27,32)(33,36,39,34)(35,40,37,42)(38,41,46,44)(43,48,45,47);(1,5,7,2)(3,4,10,8)(6,9,15,13)(11,14,16,12)(17,20)(18,26)(19,29)(21,23)(22,27)(24,31)(25,30)(28,32)(33,37,39,35)(34,40,36,42)(38,43,46,45)(41,48,44,47);(1,6,7,15)(2,9,5,13)(3,11,10,16)(4,12,8,14)(17,21,22,26)(18,27,23,20)(19,30,28,31)(24,32,25,29)(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=h,d^2=gh,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, 8633580);