#408175

order	512
nilpotency class	3
isoclinism	isoclinic class of #396118
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂²×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₂×C₄	#555
commutator, Frattini, cosocle	C₂³×C₄	C₂⁴
socle	C₂²	#2420
prime radical	C₂²×#41	C₂³
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₄ ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

732 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)(3,6)(4,8)(5,7)(9,10)(11,23)(12,19)(13,18)(14,21)(15,20)(16,22)(17,25)(24,31)(26,34)(27,33)(28,35)(29,37)(30,36)(32,40)(38,39);(1,3,5,4)(2,6,7,8)(9,11)(10,17)(12,28)(13,24)(14,26)(15,25)(16,27)(18,35)(19,32)(20,33)(21,23)(22,34)(29,40)(30,39)(31,36)(37,38);(2,7)(6,8)(9,12,14,13,16,30,15,29)(10,18,20,19,22,37,21,36)(11,24,25,28,27,40,26,39)(17,31,23,38,34,32,33,35);(1,4,5,3)(2,8,7,6)(9,13,14,30,16,29,15,12)(10,19,20,37,22,36,21,18)(11,24,26,39,27,40,25,28)(17,32,33,38,34,31,23,35)

Polycyclic presentation:

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

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, 315371);