#7927649

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

4622 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)(11,15)(12,20)(13,17)(14,18)(16,22)(19,24)(21,23)(25,26,30,34)(27,31,36,40)(28,37,38,32)(29,33,35,39);(1,3,5,8)(2,6,4,7)(9,11,14,13)(10,15,18,17)(12,19,22,21)(16,23,20,24)(25,27,30,36)(26,31,34,40)(28,35,38,29)(32,33,37,39);(9,10,14,18)(11,15,13,17)(12,16,22,20)(19,23,21,24)(25,28,30,38)(26,32,34,37)(27,29,36,35)(31,39,40,33);(1,4,5,2)(3,7,8,6)(9,10)(11,15)(12,16)(13,17)(14,18)(19,23)(20,22)(21,24)(25,28)(26,32)(27,35)(29,36)(30,38)(31,33)(34,37)(39,40);(9,12)(10,16)(11,19)(13,21)(14,22)(15,23)(17,24)(18,20)(25,26)(27,31)(28,32)(29,39)(30,34)(33,35)(36,40)(37,38);(9,13,14,11)(10,17,18,15)(12,21,22,19)(16,24,20,23)(25,29,30,35)(26,33,34,39)(27,28,36,38)(31,37,40,32)

Polycyclic presentation:

2^9|a^2=gi,b^2=gh,c^2=h,d^2=g,f^2=h,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, 7834845);