#7729196

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	#35210	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4297 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)(11,19)(12,18)(13,16)(14,20)(15,17)(21,22)(23,24);(1,2)(3,6)(4,5)(7,8)(9,11,15,13)(10,16,17,19)(12,21,20,23)(14,22,18,24)(25,26)(27,32)(28,33)(29,37)(30,31)(34,38)(35,36)(39,40);(1,3,4,7)(2,6,5,8)(10,17)(12,20)(16,19)(21,23)(26,31)(28,32)(34,37)(36,39);(9,12,15,20)(10,18,17,14)(11,21,13,23)(16,24,19,22)(25,27,30,33)(26,32,31,28)(29,35,38,40)(34,39,37,36);(9,13,15,11)(10,19,17,16)(12,23,20,21)(14,24,18,22)(25,28,30,32)(26,33,31,27)(29,36,38,39)(34,40,37,35);(9,14,15,18)(10,20,17,12)(11,22,13,24)(16,23,19,21)(25,29)(26,34)(27,35)(28,36)(30,38)(31,37)(32,39)(33,40)

Polycyclic presentation:

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

Conjugacy classes & character table

98 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 7636392);