#7715945

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×#534	C₂²
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4478 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)(9,10,15,12)(11,17,13,16)(14,20,22,18)(19,23,21,24)(25,26,31,28)(27,35,36,32)(29,33,40,37)(30,38,39,34);(1,3,5,8)(2,6,4,7)(9,11)(10,16)(12,17)(13,15)(14,19)(18,23)(20,24)(21,22)(25,27,31,36)(26,32,28,35)(29,39,40,30)(33,38,37,34);(10,12)(11,13)(18,20)(19,21)(27,36)(29,40)(32,35)(33,37);(1,4,5,2)(3,7,8,6)(9,12,15,10)(11,17,13,16)(14,20,22,18)(19,24,21,23)(25,28,31,26)(27,35,36,32)(29,37,40,33)(30,38,39,34);(9,13,15,11)(10,17,12,16)(14,21,22,19)(18,24,20,23)(25,29)(26,33)(27,30)(28,37)(31,40)(32,34)(35,38)(36,39);(9,14)(10,18)(11,19)(12,20)(13,21)(15,22)(16,23)(17,24)(25,30)(26,34)(27,29)(28,38)(31,39)(32,33)(35,37)(36,40)

Polycyclic presentation:

2^9|a^2=g,b^2=gh,d^2=g,e^2=h,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, 7623141);