#7658654

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	#37587	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4317 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)(9,10,15,16)(11,17,21,24)(12,14,20,18)(13,19,22,23)(25,26)(27,31)(28,37)(29,32)(30,33)(34,40)(35,38)(36,39);(1,2)(3,5)(4,6)(7,8)(10,16)(11,21)(12,20)(19,23)(25,27,30,36)(26,31,33,39)(28,34,32,38)(29,35,37,40);(9,11)(10,17)(12,22)(13,20)(14,23)(15,21)(16,24)(18,19)(25,28,30,32)(26,29,33,37)(27,34,36,38)(31,35,39,40);(1,3,4,7)(2,5,6,8)(9,12,15,20)(10,18,16,14)(11,22,21,13)(17,19,24,23)(25,27)(26,31)(28,34)(29,35)(30,36)(32,38)(33,39)(37,40);(9,13,15,22)(10,19,16,23)(11,20,21,12)(14,17,18,24)(25,26)(27,31)(28,29)(30,33)(32,37)(34,35)(36,39)(38,40);(9,14,15,18)(10,20,16,12)(11,23,21,19)(13,24,22,17)(25,29)(26,32)(27,35)(28,33)(30,37)(31,38)(34,39)(36,40)

Polycyclic presentation:

2^9|a^2=g,b^2=h,c^2=h,d^2=g,e^2=i,f^2=i,b^a=bg,c^a=ch,d^a=di,c^b=ci,d^b=dhi,f^e=fhi

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, 7565850);