#8158863

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	#34589	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4009 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)(18,20)(21,29)(23,27)(25,31)(33,34)(35,41)(36,43)(37,45)(38,40)(39,46)(42,44)(47,48);(1,3,6,8)(2,4,7,12)(5,10,14,15)(9,11,13,16)(17,18,22,20)(19,23,26,27)(21,24,29,28)(25,30,31,32)(33,34,38,40)(35,41,43,36)(37,39,46,45)(42,47,48,44);(2,7)(3,8)(9,13)(10,15)(17,19)(18,23)(20,27)(21,25)(22,26)(24,30)(28,32)(29,31)(33,34,38,40)(35,41,43,36)(37,39,46,45)(42,47,48,44);(1,3)(2,4)(5,10)(6,8)(7,12)(9,11)(13,16)(14,15)(17,20,22,18)(19,23,26,27)(21,28,29,24)(25,30,31,32)(33,35,38,43)(34,36,40,41)(37,42,46,48)(39,44,45,47);(1,4)(2,8)(3,7)(5,11)(6,12)(9,15)(10,13)(14,16)(17,19)(18,23)(20,27)(21,25)(22,26)(24,30)(28,32)(29,31)(33,36)(34,35)(37,44)(38,41)(39,42)(40,43)(45,48)(46,47);(1,5,6,14)(2,9,7,13)(3,10,8,15)(4,11,12,16)(17,21)(18,24)(19,25)(20,28)(22,29)(23,30)(26,31)(27,32)(33,37,38,46)(34,39,40,45)(35,42,43,48)(36,44,41,47)

Polycyclic presentation:

2^9|b^2=gi,c^2=hi,d^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, 8066059);