#405454

order	512
nilpotency class	3
isoclinism	isoclinic class of #394746
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁶

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₂×C₄	C₂×#124
commutator, Frattini, cosocle	C₂³×C₄	C₂⁴
socle	C₂²	C₂×#394
prime radical	#60444	C₂
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₄ ◁ C₂⁴×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

1516 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)(12,25)(13,20)(14,19)(15,18)(16,23)(17,27)(22,32)(24,35)(26,33)(28,34)(29,31)(30,38)(36,37)(39,40);(1,3,4,7)(2,5,6,8)(9,11,14,29,15,24,13,23)(10,16,20,35,18,31,19,21)(12,22,27,39,28,38,26,37)(17,32,25,36,33,30,34,40);(9,12,13,26,15,28,14,27)(10,17,19,34,18,33,20,25)(11,22,23,37,24,38,29,39)(16,30,21,36,31,32,35,40);(2,6)(5,8)(10,18)(16,31)(17,33)(19,20)(21,35)(25,34)(30,32)(36,40)

Polycyclic presentation:

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

Conjugacy classes & character table

80 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 312650);