#503483

order	512
nilpotency class	4
isoclinism	isoclinic class of #501132
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁴×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	C₂×B(4,𝔽₂)
commutator, Frattini, cosocle	C₂²×Q₈	C₂⁴
prime radical	C₂×#555	C₂²
LCS	G ▷ C₂²×Q₈ ▷ C₂² ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂²×C₄ ◁ C₂×C₄×Q₈ ◁ G
LECS	G ▷ C₂²×Q₈ ▷ C₂² ▷ C₂ ▷ C₁

900 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)(3,7)(4,6)(5,8)(9,10,13,16)(11,21,25,38)(12,20,15,18)(14,28,23,19)(17,31,30,26)(22,37,24,36)(27,34,29,35)(32,40,33,39);(1,3,5,4)(2,6,8,7)(10,16)(11,22)(12,15)(17,32)(19,28)(21,36)(24,25)(26,40)(27,29)(30,33)(31,39)(37,38);(1,3,5,4)(2,7,8,6)(9,11,23,31,13,25,14,26)(10,17,28,38,16,30,19,21)(12,24,29,39,15,22,27,40)(18,32,35,36,20,33,34,37);(1,4,5,3)(2,7,8,6)(9,12,13,15)(10,18,16,20)(11,24,25,22)(14,27,23,29)(17,32,30,33)(19,34,28,35)(21,37,38,36)(26,40,31,39)

Polycyclic presentation:

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

Conjugacy classes & character table

47 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 410679);