#7732309

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	#37269	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4617 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)(9,10,15,17)(11,20,12,16)(13,14,19,18)(21,22,23,24)(25,26,31,30)(27,35,36,32)(28,33,38,39)(29,34,37,40);(1,2)(3,6)(4,5)(7,8)(9,11,15,12)(10,16,17,20)(13,21,19,23)(14,24,18,22)(25,27)(26,32)(28,37)(29,38)(30,35)(31,36)(33,34)(39,40);(10,17)(11,12)(13,19)(22,24)(27,36)(29,37)(32,35)(34,40);(1,3,4,7)(2,6,5,8)(9,12,15,11)(10,16,17,20)(13,23,19,21)(14,24,18,22)(25,28,31,38)(26,33,30,39)(27,37,36,29)(32,34,35,40);(9,13,15,19)(10,18,17,14)(11,21,12,23)(16,22,20,24)(25,29,31,37)(26,34,30,40)(27,38,36,28)(32,33,35,39);(9,14,15,18)(10,19,17,13)(11,22,12,24)(16,21,20,23)(25,30,31,26)(27,32,36,35)(28,39,38,33)(29,40,37,34)

Polycyclic presentation:

2^9|a^2=i,b^2=h,d^2=g,e^2=i,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, 7639505);