#7644747

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	C₂³×D₄	C₂³
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4558 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:

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

Polycyclic presentation:

2^9|a^2=g,b^2=h,c^2=gh,d^2=g,b^a=bg,c^a=ch,d^a=dh,f^a=fhi,c^b=ci,e^d=ehi

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, 7551943);