#7826658

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	#37129	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4682 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)(11,15)(12,19)(13,14)(16,21)(17,22)(18,24)(20,23);(1,2)(3,5)(4,6)(7,8)(9,10,13,14)(11,15,17,22)(12,16,21,19)(18,23,20,24)(25,26)(27,32)(28,29)(30,31)(33,38)(34,40)(35,37)(36,39);(1,3,4,7)(2,5,6,8)(10,14)(11,17)(16,19)(18,20)(26,31)(29,38)(32,39)(35,40);(9,11)(10,15)(12,20)(13,17)(14,22)(16,24)(18,21)(19,23)(25,27)(26,32)(28,37)(29,35)(30,36)(31,39)(33,34)(38,40);(9,12,13,21)(10,16,14,19)(11,18,17,20)(15,23,22,24)(25,28,30,33)(26,29,31,38)(27,34,36,37)(32,40,39,35);(9,10)(11,15)(12,16)(13,14)(17,22)(18,23)(19,21)(20,24)(25,29)(26,33)(27,35)(28,31)(30,38)(32,34)(36,40)(37,39)

Polycyclic presentation:

2^9|a^2=gh,b^2=h,c^2=gh,e^2=i,b^a=bg,c^a=ch,e^a=eh,c^b=ci,f^b=fi,d^c=dh,f^c=fi,e^d=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, 7733854);