#8159027

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	#56800	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

3971 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)(3,4)(5,7)(6,13)(8,10)(9,14)(11,16)(12,15)(18,20)(19,25)(21,29)(30,32)(33,34)(35,42)(36,37)(38,40)(39,43)(41,48)(44,47)(45,46);(1,3,7,8)(2,4,5,10)(6,11,14,15)(9,12,13,16)(17,18,22,20)(19,26,25,23)(21,24,29,28)(27,32,31,30)(33,35,39,38)(34,40,43,42)(36,44,41,46)(37,45,48,47);(1,4,7,10)(2,8,5,3)(6,12,14,16)(9,15,13,11)(17,18,22,20)(19,26,25,23)(21,24,29,28)(27,32,31,30)(33,36,39,41)(34,37,43,48)(35,44,38,46)(40,45,42,47);(1,4)(2,3)(5,8)(6,12)(7,10)(9,11)(13,15)(14,16)(17,19,22,25)(18,23,20,26)(21,27,29,31)(24,30,28,32)(33,37,39,48)(34,41,43,36)(35,45,38,47)(40,46,42,44);(1,5,7,2)(3,10,8,4)(6,13,14,9)(11,16,15,12)(17,20,22,18)(19,23,25,26)(21,28,29,24)(27,30,31,32)(33,34)(35,40)(36,37)(38,42)(39,43)(41,48)(44,45)(46,47);(1,6,7,14)(2,9,5,13)(3,11,8,15)(4,12,10,16)(17,21)(18,24)(19,27)(20,28)(22,29)(23,30)(25,31)(26,32)(33,38,39,35)(34,42,43,40)(36,46,41,44)(37,47,48,45)

Polycyclic presentation:

2^9|b^2=gi,c^2=gi,d^2=g,e^2=h,f^2=gh,b^a=bg,c^a=ch,d^a=dg,e^a=eh,f^a=fgi,c^b=ci,d^b=dh,d^c=dgi,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, 8066223);