#8186899

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	#34964	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4445 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 40
transitive degree	μ_t(G) = 256

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

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

Polycyclic presentation:

2^9|a^2=hi,b^2=g,c^2=h,d^2=gi,e^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, 8094095);