#499439

order	512
nilpotency class	4
isoclinism	isoclinic class of #13371
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁴

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₂×C₄	#476
commutator, Frattini, cosocle	C₂²×C₈	C₂⁴
socle	C₂²	C₂×#476
prime radical	C₂⁴	AGL(1,ℤ/8)
LCS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₄ ◁ C₂³×C₄ ◁ C₂²×C₄×C₈ ◁ G
LECS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁

600 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 36
transitive degree	μ_t(G) = 64

A minimal faithful permutation representation — 4 generators on 36 points:

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

Polycyclic presentation:

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

Conjugacy classes & character table

80 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 406635);