#8184829

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	#34957	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4548 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 40
transitive degree	μ_t(G) = 128

A minimal faithful permutation representation — 6 generators on 40 points:

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

Polycyclic presentation:

2^9|a^2=ghi,b^2=gi,c^2=gh,d^2=h,e^2=h,f^2=h,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, 8092025);