#7710859

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	#40006	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4391 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)(12,20)(13,15)(14,18)(16,21)(19,22)(23,24)(29,37)(33,38)(34,39)(36,40);(1,2)(3,5)(4,6)(7,8)(9,11)(10,14)(12,19)(13,18)(15,17)(16,22)(20,23)(21,24)(25,26,30,31)(27,28,32,35)(29,33,37,38)(34,36,39,40);(10,15)(11,18)(16,20)(19,24)(26,31)(28,35)(33,38)(36,40);(9,10)(11,14)(12,16)(13,15)(17,18)(19,22)(20,21)(23,24)(25,27,30,32)(26,28,31,35)(29,34,37,39)(33,36,38,40);(1,3,4,7)(2,5,6,8)(9,11)(10,14)(12,19)(13,18)(15,17)(16,22)(20,23)(21,24)(25,28)(26,32)(27,31)(29,36)(30,35)(33,39)(34,38)(37,40);(9,12,13,21)(10,16,15,20)(11,19,18,24)(14,22,17,23)(25,29)(26,33)(27,34)(28,36)(30,37)(31,38)(32,39)(35,40)

Polycyclic presentation:

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

Conjugacy classes & character table

104 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 7618055);