#7732376

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	#35534	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4480 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 40
transitive degree	μ_t(G) = 256

A minimal faithful permutation representation — 6 generators on 40 points:

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

Polycyclic presentation:

2^9|a^2=ghi,d^2=g,e^2=hi,f^2=i,b^a=bg,c^a=ch,e^a=eh,c^b=ci,f^b=fi,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, 7639572);