#7798047

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
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4528 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:

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

Polycyclic presentation:

2^9|a^2=hi,b^2=ghi,c^2=gh,d^2=i,e^2=i,f^2=h,b^a=bg,c^a=ch,e^a=eh,c^b=ci,e^b=ei,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, 7705243);