#8718120

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

4028 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:

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

Polycyclic presentation:

2^9|a^2=gi,b^2=ghi,c^2=h,d^2=gi,e^2=g,f^2=h,b^a=bg,c^a=ch,d^a=dg,e^a=eh,c^b=ci,d^b=dh,d^c=dgi,f^d=fgi,f^e=fgi

Conjugacy classes & character table

89 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 8625316);