#358054

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

1818 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:

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

Polycyclic presentation:

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

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, 265250);