#7915873

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	#2410	C₂²
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4455 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)(9,10,16,22)(11,15,21,17)(12,20,14,18)(13,24,23,19)(25,26,29,32)(27,30,35,39)(28,31,36,40)(33,37,34,38);(1,2)(3,6)(4,5)(7,8)(9,11)(10,17)(12,13)(14,23)(15,22)(16,21)(18,24)(19,20)(25,27)(26,30)(28,34)(29,35)(31,38)(32,39)(33,36)(37,40);(1,3,4,7)(2,6,5,8)(9,12,16,14)(10,18,22,20)(11,23,21,13)(15,24,17,19)(25,28)(26,31)(27,33)(29,36)(30,37)(32,40)(34,35)(38,39);(9,13,16,23)(10,19,22,24)(11,12,21,14)(15,18,17,20)(25,26)(27,30)(28,31)(29,32)(33,37)(34,38)(35,39)(36,40);(9,14,16,12)(10,20,22,18)(11,23,21,13)(15,24,17,19)(25,28)(26,31)(27,34)(29,36)(30,38)(32,40)(33,35)(37,39);(9,15)(10,21)(11,22)(12,24)(13,18)(14,19)(16,17)(20,23)(25,27)(26,30)(28,33)(29,35)(31,37)(32,39)(34,36)(38,40)

Polycyclic presentation:

2^9|a^2=i,c^2=g,d^2=h,e^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, 7823069);