#8110446

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	#40420	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4056 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)(3,10)(4,7)(5,12)(6,8)(9,14)(11,15)(13,16)(18,22)(23,28)(24,29)(30,32)(37,45)(42,46)(43,48)(44,47);(1,3,7,6)(2,8,4,10)(5,11,14,13)(9,15,12,16)(19,25)(23,28)(26,31)(30,32)(34,39)(36,40)(42,46)(44,47);(1,2,7,4)(3,10,6,8)(5,9,14,12)(11,16,13,15)(17,18)(19,23)(20,24)(21,22)(25,28)(26,30)(27,29)(31,32)(33,34)(35,40)(36,41)(37,42)(38,39)(43,47)(44,48)(45,46);(1,4,7,2)(3,8,6,10)(5,12,14,9)(11,15,13,16)(17,19,21,25)(18,23,22,28)(20,26,27,31)(24,30,29,32)(33,35,38,41)(34,40,39,36)(37,43,45,48)(42,47,46,44);(1,5)(2,9)(3,11)(4,12)(6,13)(7,14)(8,15)(10,16)(17,18)(19,23)(20,24)(21,22)(25,28)(26,30)(27,29)(31,32)(33,36,38,40)(34,41,39,35)(37,44,45,47)(42,48,46,43);(1,6,7,3)(2,8,4,10)(5,13,14,11)(9,15,12,16)(17,20,21,27)(18,24,22,29)(19,26,25,31)(23,30,28,32)(33,37)(34,42)(35,43)(36,44)(38,45)(39,46)(40,47)(41,48)

Polycyclic presentation:

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

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, 8017642);