#8197261

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	#54500	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4048 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

92 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 8104457);