#8696302

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

4176 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:

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

Polycyclic presentation:

2^9|a^2=ghi,c^2=i,d^2=h,e^2=gh,f^2=g,b^a=bg,c^a=ch,d^a=dg,e^a=eh,c^b=ci,d^b=dghi,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, 8603498);