#8749758

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	#2416	C₂²
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

3827 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)(17,18,21,22)(19,25,26,23)(20,28,29,24)(27,30,32,31)(34,36)(39,41)(40,44)(46,47);(1,3,7,6)(2,8,9,10)(4,13,14,11)(5,12,15,16)(18,22)(19,26)(24,28)(27,32)(33,34,38,36)(35,41,43,39)(37,40,45,44)(42,47,48,46);(2,9)(4,14)(8,10)(11,13)(17,19,21,26)(18,23,22,25)(20,27,29,32)(24,30,28,31)(33,35,38,43)(34,39,36,41)(37,42,45,48)(40,46,44,47);(1,4)(2,5)(3,11)(6,13)(7,14)(8,16)(9,15)(10,12)(17,18)(19,23)(20,24)(21,22)(25,26)(27,30)(28,29)(31,32)(33,36,38,34)(35,41,43,39)(37,44,45,40)(42,47,48,46);(1,5)(2,4)(3,12)(6,16)(7,15)(8,13)(9,14)(10,11)(17,20)(18,24)(19,27)(21,29)(22,28)(23,30)(25,31)(26,32)(33,37,38,45)(34,40,36,44)(35,42,43,48)(39,46,41,47);(1,6,7,3)(2,8,9,10)(4,11,14,13)(5,12,15,16)

Polycyclic presentation:

2^9|a^2=h,b^2=hi,c^2=i,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, 8656954);