#7719082

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	C₂×#121	C₂³
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4143 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:

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

Polycyclic presentation:

2^9|a^2=g,b^2=ghi,c^2=hi,d^2=h,e^2=ghi,f^2=g,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, 7626278);