#354955

order	512
nilpotency class	3
isoclinism	isoclinic class of #471
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄² · Schur multiplier (H₂) = C₂⁵×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	C₂²×#41	C₂³
LCS	G ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₈ ◁ C₂³×C₈ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

1504 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)(7,17,20,27)(8,14)(9,12)(10,15)(11,16)(13,19,28,24)(18,31,33,35)(21,25)(22,29)(23,30)(26,32,36,34);(6,12,16,15)(13,17,28,27)(14,25,30,29)(26,31,36,35);(2,4)(5,7)(6,13)(8,18)(9,19)(10,24)(11,20)(12,17)(14,26)(15,27)(16,28)(21,32)(22,34)(23,33)(25,31)(29,35)(30,36);(5,8,9,21,11,23,10,22)(6,14,12,25,16,30,15,29)(7,18,19,32,20,33,24,34)(13,26,17,31,28,36,27,35)

Polycyclic presentation:

2^9|a^2=g,b^2=h,d^2=eh,e^2=i,f^2=i,b^a=be,c^a=cf,e^a=ei,h^a=hi,f^c=fi,g^c=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, 262151);