#354711

order	512
nilpotency class	3
isoclinism	isoclinic class of #471
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄² · Schur multiplier (H₂) = C₂⁴×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₂×C₈	#121
commutator	C₂×C₄	C₂²×C₄²
Frattini, cosocle	C₂³×C₄	C₂⁴
socle	C₂²	C₄×#121
prime radical	C₂⁴	C₂×#41
LCS	G ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₈ ◁ C₂³×C₈ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

1044 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

128 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 261907);