#8270538

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	#42982	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4084 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 48
transitive degree	μ_t(G) = 128

A minimal faithful permutation representation — 6 generators on 48 points:

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

Polycyclic presentation:

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

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, 8177734);