#353522

order	512
nilpotency class	3
isoclinism	isoclinic class of #20671
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄² · Schur multiplier (H₂) = C₂⁶×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₂×C₄	C₂×#117
commutator	C₂×C₄	C₂²×C₄²
Frattini, cosocle	C₂³×C₄	C₂⁴
socle	C₂²	C₂×#380
prime radical	C₂⁵	C₂×Q₈
LCS	G ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₄ ◁ C₂³×M₁₆ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

1734 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:

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

Polycyclic presentation:

2^9|a^2=g,b^2=h,c^2=e,e^2=i,f^2=i,g^2=i,h^2=i,b^a=be,c^a=cf,e^a=ei,h^a=hi,d^b=di,g^c=gi

Conjugacy classes & character table

116 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 260718);