#7717679

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	#40381	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4423 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:

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

Polycyclic presentation:

2^9|d^2=i,e^2=g,f^2=gh,b^a=bg,c^a=ch,e^a=eh,f^a=fi,c^b=ci,e^b=ei,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, 7624875);