#8416845

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	#53440	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4039 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:

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

Polycyclic presentation:

2^9|a^2=gh,b^2=g,c^2=g,d^2=ghi,e^2=gi,f^2=h,b^a=bg,c^a=ch,d^a=di,e^a=eh,c^b=ci,d^b=dghi,e^b=egi,f^c=fgi,e^d=egi

Conjugacy classes & character table

80 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 8324041);