#7642895

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	#35190	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4505 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:

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

Polycyclic presentation:

2^9|a^2=g,b^2=gh,c^2=g,d^2=hi,e^2=gh,f^2=h,b^a=bg,c^a=ch,d^a=dh,f^a=fhi,c^b=ci,e^c=ehi

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, 7550091);