#357921

order	512
nilpotency class	3
isoclinism	isoclinic class of #2018
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂⁸

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	C₂³×#41	C₂²
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂²×C₄ ◁ C₂⁴×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

2538 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)(5,6,7,8)(9,10)(11,21)(12,26)(13,18)(14,19)(15,20)(16,24)(17,29)(22,31)(23,33)(25,32)(27,39)(28,35)(30,34)(36,37)(38,40);(9,11)(10,16)(12,27)(13,23)(14,24)(15,25)(17,31)(18,21)(19,32)(20,33)(22,30)(26,36)(28,38)(29,37)(34,40)(35,39);(2,4)(9,12,13,28,15,30,14,29)(10,17,19,34,20,35,18,26)(11,22,24,38,25,27,23,37)(16,31,21,36,33,39,32,40);(9,13,15,14)(10,18,20,19)(11,23,25,24)(12,28,30,29)(16,21,33,32)(17,26,35,34)(22,37,27,38)(31,36,39,40)

Polycyclic presentation:

2^9|a^2=h,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, 265117);