#6372969

order	512
nilpotency class	3
isoclinism	isoclinic class of #60460
automorphism	?
homology	Abelianization (H₁) = C₂⁴×C₄ · Schur multiplier (H₂) = C₂⁹

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	C₂³×D₄
commutator	C₂×C₄	C₂⁴×C₄
Frattini, cosocle	C₂²×C₄	C₂⁵
prime radical	C₂²×#41	C₂³
LCS	G ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂×C₄×#41 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

2698 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 40
transitive degree	μ_t(G) = 256

A minimal faithful permutation representation — 5 generators on 40 points:

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

Polycyclic presentation:

2^9|a^2=h,b^2=fi,d^2=hi,e^2=i,f^2=i,b^a=bf,c^a=cg,f^a=fi,c^b=ci,e^c=ei,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, 6280165);