#7630833

order	512
nilpotency class	2
automorphism	?
homology	Abelianization (H₁) = C₂⁶ · Schur multiplier (H₂) = C₂¹⁴

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, commutator, Frattini, socle, cosocle	C₂³	C₂⁶
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

10606 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:

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

Polycyclic presentation:

2^9|b^2=g,d^2=i,e^2=i,b^a=bg,c^a=ch,f^a=fi,c^b=ci,e^d=ei

Conjugacy classes & character table

116 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 7538029);