#8096089

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	#2537	C₂²
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4026 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:

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

Polycyclic presentation:

2^9|a^2=h,c^2=g,d^2=g,e^2=i,f^2=i,b^a=bg,c^a=ch,d^a=dg,c^b=ci,d^b=dh,f^e=fghi

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, 8003285);