#7844552

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

4349 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)(9,10,14,15)(11,16,17,12)(13,18,22,23)(19,24,21,20)(25,26,31,35)(27,32,38,40)(28,33,36,37)(29,34,39,30);(1,3,5,8)(2,6,4,7)(9,11)(10,12)(13,21)(14,17)(15,16)(18,24)(19,22)(20,23)(25,27)(26,32)(28,39)(29,36)(30,33)(31,38)(34,37)(35,40);(10,15)(11,17)(13,22)(20,24)(25,28,31,36)(26,33,35,37)(27,29,38,39)(30,32,34,40);(1,4,5,2)(3,7,8,6)(9,11)(10,16)(12,15)(13,19)(14,17)(18,24)(20,23)(21,22)(25,27)(26,32)(28,29)(30,37)(31,38)(33,34)(35,40)(36,39);(9,12,14,16)(10,17,15,11)(13,20,22,24)(18,21,23,19)(25,29)(26,34)(27,36)(28,38)(30,35)(31,39)(32,37)(33,40);(9,13,14,22)(10,18,15,23)(11,19,17,21)(12,20,16,24)(25,30,31,34)(26,29,35,39)(27,37,38,33)(28,40,36,32)

Polycyclic presentation:

2^9|a^2=ghi,b^2=gh,c^2=hi,d^2=gh,e^2=h,f^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, 7751748);