#8119554

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	#56114	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

3929 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

86 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 8026750);