#7720320

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	#35261	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4413 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)(9,10,13,14)(11,15,17,21)(12,19,20,16)(18,24,23,22)(25,26,31,28)(27,35,37,32)(29,33,40,38)(30,39,36,34);(1,2)(3,7)(4,8)(5,6)(9,10,13,14)(11,15,17,21)(12,16,20,19)(18,22,23,24)(25,27)(26,32)(28,35)(29,30)(31,37)(33,34)(36,40)(38,39);(1,3,5,4)(2,7,6,8)(10,14)(11,17)(16,19)(18,23)(25,26,31,28)(27,35,37,32)(29,38,40,33)(30,34,36,39);(1,4,5,3)(2,8,6,7)(9,11)(10,15)(12,18)(13,17)(14,21)(16,22)(19,24)(20,23)(25,28,31,26)(27,35,37,32)(29,38,40,33)(30,39,36,34);(9,10)(11,15)(12,16)(13,14)(17,21)(18,22)(19,20)(23,24)(25,29)(26,33)(27,36)(28,38)(30,37)(31,40)(32,39)(34,35);(9,12,13,20)(10,16,14,19)(11,18,17,23)(15,22,21,24)(25,30,31,36)(26,34,28,39)(27,29,37,40)(32,33,35,38)

Polycyclic presentation:

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

Conjugacy classes & character table

98 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 7627516);