#358336

order	512
nilpotency class	3
isoclinism	isoclinic class of #2018
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂⁷

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₂²×C₄	#121
commutator	C₂²×C₄	C₂³×C₄
Frattini, cosocle	C₂³×C₄	C₂⁴
socle	C₂³	C₂×#121
prime radical	C₂⁵	#41
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂²×C₄ ◁ C₂⁴×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

1502 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 40
transitive degree	μ_t(G) = 128

A minimal faithful permutation representation — 4 generators on 40 points:

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

Polycyclic presentation:

2^9|a^2=h,b^2=ef,c^2=e,d^2=i,e^2=i,f^2=i,b^a=be,c^a=cf,e^a=ei,f^a=fi,g^a=gi,c^b=cg,e^b=ei,f^b=fi,g^c=gi

Conjugacy classes & character table

104 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 265532);