#7842450

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	#50688	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4366 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:

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

Polycyclic presentation:

2^9|a^2=ghi,b^2=hi,c^2=h,d^2=gh,e^2=gh,f^2=gi,b^a=bg,c^a=ch,e^a=eh,c^b=ci,f^b=fi,d^c=dh,f^c=fi,e^d=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, 7749646);