#497851

order	512
nilpotency class	4
isoclinism	isoclinic class of #33830
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂⁵

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₂×C₄	C₂²×D₈
commutator	C₂×C₈	C₂³×C₄
Frattini, cosocle	C₂²×C₈	C₂⁴
socle	C₂²	C₂²×#107
prime radical	C₂²×D₄	Q₁₆
LCS	G ▷ C₂×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₄ ◁ C₂×C₄×D₄ ◁ C₂×C₈×D₄ ◁ G
LECS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁

606 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 36
transitive degree	μ_t(G) = 128

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

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

Polycyclic presentation:

2^9|a^2=g,c^2=fh,f^2=hi,g^2=i,h^2=i,b^a=be,c^a=cf,f^a=fh,h^a=hi,c^b=ci,d^b=de,d^c=di

Conjugacy classes & character table

86 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 405047);