#8158880

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	#54639	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

3984 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)(5,13)(6,7)(8,12)(9,14)(10,16)(11,15)(17,18,23,21)(19,22,27,24)(20,28,29,25)(26,31,32,30)(33,34,39,42)(35,36,37,40)(38,46,47,41)(43,48,45,44);(1,3,6,8)(2,4,7,12)(5,10,14,15)(9,11,13,16)(17,19,23,27)(18,24,21,22)(20,26,29,32)(25,31,28,30)(33,35,39,37)(34,40,42,36)(38,43,47,45)(41,48,46,44);(2,7)(3,8)(9,13)(10,15)(20,29)(25,28)(26,32)(30,31)(33,35,39,37)(34,40,42,36)(38,43,47,45)(41,48,46,44);(1,3)(2,4)(5,10)(6,8)(7,12)(9,11)(13,16)(14,15)(17,20,23,29)(18,25,21,28)(19,26,27,32)(22,30,24,31)(33,36,39,40)(34,35,42,37)(38,44,47,48)(41,43,46,45);(1,4)(2,8)(3,7)(5,11)(6,12)(9,15)(10,13)(14,16)(17,21,23,18)(19,22,27,24)(20,25,29,28)(26,31,32,30)(33,37,39,35)(34,36,42,40)(38,45,47,43)(41,44,46,48);(1,5,6,14)(2,9,7,13)(3,10,8,15)(4,11,12,16)(17,22)(18,19)(20,30)(21,27)(23,24)(25,26)(28,32)(29,31)(33,38,39,47)(34,41,42,46)(35,43,37,45)(36,44,40,48)

Polycyclic presentation:

2^9|a^2=g,b^2=gi,c^2=hi,d^2=g,e^2=g,f^2=h,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, 8066076);