#7972322

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

4295 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

101 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 7879518);