#8125490

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	#57215	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

3994 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)(3,8,5,4)(6,14,15,10)(11,12,13,16)(17,18,23,20)(19,24,26,21)(22,28,30,25)(27,29,32,31)(33,34,40,39)(35,44,38,41)(36,43,37,42)(45,47,46,48);(1,3,7,5)(2,8,9,4)(6,11,15,13)(10,16,14,12)(17,19,23,26)(18,24,20,21)(22,27,30,32)(25,31,28,29)(33,35,40,38)(34,41,39,44)(36,46,37,45)(42,48,43,47);(2,9)(3,5)(10,14)(11,13)(19,26)(21,24)(27,32)(29,31)(33,36,40,37)(34,42,39,43)(35,45,38,46)(41,47,44,48);(1,4,7,8)(2,3,9,5)(6,12,15,16)(10,11,14,13)(17,20,23,18)(19,21,26,24)(22,28,30,25)(27,29,32,31)(33,37,40,36)(34,43,39,42)(35,46,38,45)(41,48,44,47);(1,5,7,3)(2,8,9,4)(6,13,15,11)(10,16,14,12)(17,21)(18,19)(20,26)(22,29)(23,24)(25,27)(28,32)(30,31)(33,38,40,35)(34,44,39,41)(36,46,37,45)(42,48,43,47);(1,6)(2,10)(3,11)(4,12)(5,13)(7,15)(8,16)(9,14)(17,22,23,30)(18,25,20,28)(19,27,26,32)(21,29,24,31)(33,39,40,34)(35,44,38,41)(36,43,37,42)(45,48,46,47)

Polycyclic presentation:

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

Conjugacy classes & character table

86 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 8032686);