#7732071

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	#37268	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4527 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)(11,16)(12,17)(13,15)(14,18)(19,23)(20,24)(21,22)(25,26)(27,37)(28,33)(29,34)(30,35)(31,36)(32,38)(39,40);(1,2)(3,6)(4,5)(7,8)(9,11,15,12)(10,16,13,17)(14,20,22,19)(18,24,21,23)(25,27)(26,32)(28,39)(29,35)(30,40)(31,38)(33,34)(36,37);(10,13)(11,12)(18,21)(19,20)(27,38)(29,39)(32,37)(34,40);(1,3,4,7)(2,6,5,8)(9,12,15,11)(10,17,13,16)(14,20,22,19)(18,24,21,23)(25,28,31,35)(26,33,36,30)(27,39,38,29)(32,34,37,40);(9,13,15,10)(11,17,12,16)(14,21,22,18)(19,24,20,23)(25,29,31,39)(26,34,36,40)(27,35,38,28)(30,32,33,37);(9,14,15,22)(10,18,13,21)(11,19,12,20)(16,23,17,24)(25,30,31,33)(26,35,36,28)(27,34,38,40)(29,32,39,37)

Polycyclic presentation:

2^9|b^2=h,d^2=gh,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, 7639267);