#7927194

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	#40391	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4302 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)(11,20)(12,23)(13,24)(14,19)(15,17)(16,21)(18,22)(25,26,31,33)(27,36,39,32)(28,34,40,30)(29,37,38,35);(1,3,5,8)(2,6,4,7)(9,11)(10,16)(12,22)(13,19)(14,18)(15,21)(17,20)(23,24)(25,27)(26,32)(28,37)(29,34)(30,38)(31,39)(33,36)(35,40);(10,17)(11,21)(12,19)(18,24)(26,33)(28,40)(32,36)(35,37);(9,12,15,19)(10,14,17,23)(11,22,21,13)(16,18,20,24)(25,28)(26,34)(27,37)(29,32)(30,33)(31,40)(35,39)(36,38);(9,13,15,22)(10,18,17,24)(11,19,21,12)(14,20,23,16)(25,29,31,38)(26,35,33,37)(27,34,39,30)(28,32,40,36);(1,4,5,2)(3,7,8,6)(9,14)(10,19)(11,18)(12,17)(13,16)(15,23)(20,22)(21,24)(25,30,31,34)(26,28,33,40)(27,38,39,29)(32,37,36,35)

Polycyclic presentation:

2^9|a^2=gi,b^2=gh,d^2=i,e^2=h,f^2=gi,b^a=bg,c^a=ch,d^a=di,e^a=eh,c^b=ci,d^c=dh,f^e=fi

Conjugacy classes & character table

95 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 7834390);