#8371259

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	#2617	C₂²
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

3993 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:

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

Polycyclic presentation:

2^9|a^2=gi,b^2=ghi,c^2=gh,d^2=hi,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,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, 8278455);