#8696142

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	#56766	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4100 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:

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

Polycyclic presentation:

2^9|a^2=hi,c^2=ghi,e^2=h,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, 8603338);