#415457

order	512
nilpotency class	3
isoclinism	isoclinic class of #395174
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁴

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₂×C₄	#535
commutator, Frattini, cosocle	C₂³×C₄	C₂⁴
socle	C₂²	#2535
prime radical	C₂⁴	C₂×#41
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₄ ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

764 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

68 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 322653);