#8172622

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	#40007	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4013 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:

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

Polycyclic presentation:

2^9|a^2=g,b^2=h,c^2=hi,d^2=h,e^2=gh,f^2=gi,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, 8079818);