#8639782

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

4027 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)(3,5)(4,8)(6,7)(9,10)(11,15)(12,13)(14,16)(17,18,22,24)(19,26,21,23)(20,28,30,25)(27,31,29,32)(33,34)(35,40)(36,42)(37,41)(38,39)(43,47)(44,48)(45,46);(1,3,6,10)(2,5,7,9)(4,12,14,11)(8,13,16,15)(17,19,22,21)(18,23,24,26)(20,29,30,27)(25,32,28,31)(34,39)(40,42)(41,46)(47,48);(2,7)(3,10)(8,16)(11,12)(18,24)(23,26)(25,28)(31,32)(33,35,38,36)(34,40,39,42)(37,43,45,44)(41,47,46,48);(1,4)(2,8)(3,11)(5,15)(6,14)(7,16)(9,13)(10,12)(17,20,22,30)(18,25,24,28)(19,27,21,29)(23,31,26,32)(33,36,38,35)(34,40,39,42)(37,43,45,44)(41,48,46,47);(1,3)(2,9)(4,11)(5,7)(6,10)(8,13)(12,14)(15,16)(17,18)(19,26)(20,25)(21,23)(22,24)(27,32)(28,30)(29,31)(33,34)(35,42)(36,40)(37,41)(38,39)(43,48)(44,47)(45,46);(1,5,6,9)(2,3,7,10)(4,13,14,15)(8,12,16,11)(17,21,22,19)(18,23,24,26)(20,27,30,29)(25,32,28,31)(33,37,38,45)(34,41,39,46)(35,43,36,44)(40,47,42,48)

Polycyclic presentation:

2^9|a^2=hi,b^2=h,c^2=ghi,d^2=g,f^2=gi,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, 8546978);