#411285

order	512
nilpotency class	3
isoclinism	isoclinic class of #393906
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁵

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	C₂×#525	C₂²
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₄ ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

964 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)(3,8)(4,7)(5,6)(9,10)(11,21,18,13)(12,25,28,38)(14,20)(15,23)(16,19)(17,29,36,30)(22,34)(24,32,33,31)(26,27)(35,40)(37,39);(2,6)(7,8)(9,11,15,33,16,18,14,24)(10,13,23,32,19,21,20,31)(12,26,30,22,28,39,29,35)(17,34,38,37,36,40,25,27);(1,3,5,4)(2,7,6,8)(9,12)(10,17,19,36)(11,22,18,35)(13,27)(14,29)(15,30)(16,28)(20,25,23,38)(21,37)(24,26,33,39)(31,40)(32,34);(1,4,5,3)(2,7,6,8)(9,13,14,31,16,21,15,32)(10,18,20,33,19,11,23,24)(12,27,29,40,28,37,30,34)(17,35,25,39,36,22,38,26)

Polycyclic presentation:

2^9|a^2=efg,b^2=gh,c^2=fg,d^2=f,f^2=i,g^2=i,b^a=be,c^a=cf,d^a=dh,e^a=ei,g^a=gi,h^a=hi,c^b=cg,g^b=gi,h^b=hi,e^c=ei,e^d=ei

Conjugacy classes & character table

74 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 318481);