#7902525

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

4490 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)(11,19)(12,23)(13,17)(14,18)(15,22)(16,21)(20,24)(25,26)(27,38)(28,35)(29,40)(30,33)(31,36)(32,37)(34,39);(1,2)(3,6)(4,5)(7,8)(9,11)(10,15)(12,20)(13,21)(14,22)(16,24)(17,23)(18,19)(25,27)(26,31)(28,29)(30,39)(32,36)(33,40)(34,35)(37,38);(9,10,14,18)(11,15,22,19)(12,16,21,23)(13,17,20,24)(25,28,32,30)(26,33,37,35)(27,39,36,29)(31,34,38,40);(1,3,4,7)(2,6,5,8)(9,10)(11,15)(12,16)(13,17)(14,18)(19,22)(20,24)(21,23)(25,29,32,39)(26,34,37,40)(27,28,36,30)(31,35,38,33);(9,12,14,21)(10,16,18,23)(11,20,22,13)(15,24,19,17)(25,30,32,28)(26,35,37,33)(27,39,36,29)(31,34,38,40);(9,13,14,20)(10,17,18,24)(11,21,22,12)(15,23,19,16)(25,31,32,38)(26,36,37,27)(28,34,30,40)(29,35,39,33)

Polycyclic presentation:

2^9|c^2=h,d^2=ghi,e^2=h,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, 7809721);