#402078

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

825 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:

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

Polycyclic presentation:

2^9|a^2=efg,b^2=ef,d^2=e,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,g^b=gi,d^c=di,g^c=gi

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, 309274);