#501591

order	512
nilpotency class	4
isoclinism	isoclinic class of #501527
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁴

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#3109
commutator, Frattini, cosocle	C₂²×C₈	C₂⁴
prime radical	#3054	C₂²
LCS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂²×C₄ ◁ C₂×C₄×M₁₆ ◁ G
LECS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁

533 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 20
transitive degree	μ_t(G) = 64

A minimal faithful permutation representation — 4 generators on 20 points:

(5,6,9,13,20,16,8,18,10,12,19,14,11,7,15,17);(1,2)(3,4)(6,12)(7,16)(8,15)(9,19);(2,4)(5,7)(6,11)(8,17)(9,13)(10,16)(12,20)(14,19)(15,18);(5,8,11,9,10,15,20,19)(6,13,16,18,12,14,7,17)

Polycyclic presentation:

2^9|a^2=ef,d^2=eh,f^2=hi,h^2=i,b^a=be,c^a=cf,d^a=deh,e^a=ei,f^a=fi,c^b=cg,d^b=di,f^b=fi,e^c=ei,f^c=fh,h^c=hi

Conjugacy classes & character table

50 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 408787);