#7826662

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	#43198	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4457 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:

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

Polycyclic presentation:

2^9|a^2=gh,b^2=h,c^2=h,d^2=g,e^2=i,b^a=bg,c^a=ch,e^a=eh,c^b=ci,f^b=fi,d^c=dh,f^c=fi,e^d=ei

Conjugacy classes & character table

98 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 7733858);