#8158747

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	#392	C₂³
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

3874 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:

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

Polycyclic presentation:

2^9|a^2=h,b^2=i,c^2=gi,e^2=g,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, 8065943);