#7719016

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	#35081	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:

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

Polycyclic presentation:

2^9|a^2=g,b^2=ghi,c^2=gi,d^2=g,e^2=hi,b^a=bg,c^a=ch,e^a=eh,f^a=fi,c^b=ci,e^b=ei,d^c=dh,e^c=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, 7626212);