#7927174

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

4514 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)(9,10,14,15)(11,18,16,12)(13,17,22,23)(19,21,24,20)(25,26)(27,29)(28,33)(30,35)(31,36)(32,34)(37,38)(39,40);(1,3,5,8)(2,6,4,7)(9,10,14,15)(11,12,16,18)(13,17,22,23)(19,20,24,21)(25,27,31,34)(26,32,36,29)(28,35,39,37)(30,33,38,40);(10,15)(11,16)(17,23)(19,24)(25,28)(26,33)(27,37)(29,38)(30,32)(31,39)(34,35)(36,40);(9,11,14,16)(10,12,15,18)(13,19,22,24)(17,20,23,21)(25,29,31,32)(26,34,36,27)(28,38,39,30)(33,35,40,37);(1,4,5,2)(3,7,8,6)(9,12)(10,16)(11,15)(13,21)(14,18)(17,19)(20,22)(23,24)(25,30,31,38)(26,35,36,37)(27,33,34,40)(28,32,39,29);(9,13)(10,17)(11,19)(12,20)(14,22)(15,23)(16,24)(18,21)(25,26)(27,32)(28,33)(29,34)(30,37)(31,36)(35,38)(39,40)

Polycyclic presentation:

2^9|a^2=gi,b^2=gh,d^2=i,e^2=g,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, 7834370);