#7715507

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	#39118	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4151 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

92 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 7622703);