#7712444

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	#2847	C₂²
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4441 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

104 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 7619640);