#7710825

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	#37805	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4453 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:

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

Polycyclic presentation:

2^9|a^2=g,b^2=i,d^2=hi,e^2=h,f^2=gh,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, 7618021);