#8691854

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	#57283	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4080 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:

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

Polycyclic presentation:

2^9|a^2=i,b^2=g,c^2=h,d^2=g,f^2=h,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, 8599050);