#8158052

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	#42595	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4153 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:

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

Polycyclic presentation:

2^9|a^2=gh,b^2=h,c^2=hi,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, 8065248);