#353715

order	512
nilpotency class	3
isoclinism	isoclinic class of #3068
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×#121
commutator	C₂×C₄	C₂²×C₄²
Frattini, cosocle	C₂³×C₄	C₂⁴
socle	C₂²	C₂×#389
prime radical	C₂⁵	C₂×Q₈
LCS	G ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₄ ◁ C₂⁴×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

2038 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:

(1,2,3,4)(5,6,9,15,11,16,10,14)(7,17,20,28,21,27,19,12)(8,22,24,13,23,30,25,29)(18,26,34,31,32,36,33,35);(5,7,10,19,11,21,9,20)(6,12,14,27,16,28,15,17)(8,18,25,33,23,32,24,34)(13,26,22,35,29,36,30,31);(5,8,9,24,11,23,10,25)(6,13,14,22,16,29,15,30)(7,18,20,34,21,32,19,33)(12,26,27,35,28,36,17,31);(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,e^2=i,f^2=i,g^2=i,h^2=i,b^a=be,c^a=cf,e^a=ei,f^a=fi,e^b=ei,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, 260911);