#8751334

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	#2540	C₂²
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

3892 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)(3,4)(5,9)(6,7)(8,12)(10,11)(13,14)(15,16)(17,18,23,27)(19,25,21,24)(20,26,31,22)(28,29,30,32)(33,34,37,38)(35,41,42,39)(36,44,45,40)(43,46,48,47);(2,7)(3,8)(9,16)(10,13)(17,19,23,21)(18,24,27,25)(20,28,31,30)(22,29,26,32)(34,38)(35,42)(40,44)(43,48);(2,7)(3,8)(9,16)(10,13)(17,19,23,21)(18,25,27,24)(20,30,31,28)(22,29,26,32)(33,35,37,42)(34,39,38,41)(36,43,45,48)(40,46,44,47);(1,3,6,8)(2,4,7,12)(5,13,15,10)(9,14,16,11)(17,20,23,31)(18,26,27,22)(19,28,21,30)(24,32,25,29)(33,34)(35,39)(36,40)(37,38)(41,42)(43,46)(44,45)(47,48);(1,4)(2,8)(3,7)(5,14)(6,12)(9,10)(11,15)(13,16)(17,21,23,19)(18,24,27,25)(20,30,31,28)(22,29,26,32)(33,36)(34,40)(35,43)(37,45)(38,44)(39,46)(41,47)(42,48);(1,5,6,15)(2,9,7,16)(3,10,8,13)(4,11,12,14)(17,22,23,26)(18,20,27,31)(19,29,21,32)(24,28,25,30)

Polycyclic presentation:

2^9|a^2=ghi,b^2=gh,c^2=ghi,d^2=gi,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, 8658530);