#8621351

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	#2676	C₂²
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

3950 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:

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

Polycyclic presentation:

2^9|a^2=i,b^2=hi,d^2=g,e^2=ghi,f^2=g,b^a=bg,c^a=ch,d^a=dg,e^a=eh,c^b=ci,d^b=dh,e^b=egi,e^c=egi,f^d=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, 8528547);