#8374190

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	#55149	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4089 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:

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

Polycyclic presentation:

2^9|a^2=gi,b^2=h,c^2=g,d^2=gh,e^2=gh,f^2=ghi,b^a=bg,c^a=ch,d^a=dg,e^a=eh,c^b=ci,d^b=dh,e^b=egi,f^c=fgi,e^d=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, 8281386);