#241055

order	512
nilpotency class	3
isoclinism	isoclinic class of #10560
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁶

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₂×C₄	#394
commutator, Frattini, cosocle	C₂³×C₄	C₂⁴
socle	C₂²	C₂×#394
prime radical	#60741	C₂
LCS	G ▷ C₂³×C₄ ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂×C₄ ◁ C₂⁴×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂² ▷ C₁

1156 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 40
transitive degree	μ_t(G) = 64

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

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

Polycyclic presentation:

2^9|c^2=e,d^2=h,e^2=h,f^2=h,b^a=be,c^a=cf,e^a=eh,f^a=fh,g^a=gi,c^b=cg,e^b=eh,f^b=fi,g^c=gh

Conjugacy classes & character table

62 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 148251);