#8159090

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	#2678	C₂²
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

3913 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 48
transitive degree	μ_t(G) = 128

A minimal faithful permutation representation — 6 generators on 48 points:

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

Polycyclic presentation:

2^9|a^2=gh,b^2=i,c^2=gi,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, 8066286);