#354687

order	512
nilpotency class	3
isoclinism	isoclinic class of #3068
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄² · Schur multiplier (H₂) = C₂⁴×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₂×C₄	C₂×#121
commutator	C₂×C₄	C₂²×C₄²
Frattini, cosocle	C₂³×C₄	C₂⁴
socle	C₂²	C₂×#389
prime radical	C₂⁴	C₂²×Q₈
LCS	G ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₄ ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

1022 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:

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

Polycyclic presentation:

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

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, 261883);