#7935921

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×#120	C₂³
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4245 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:

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

Polycyclic presentation:

2^9|b^2=gi,c^2=hi,d^2=ghi,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, 7843117);