#354641

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

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×#135	C₂²
LCS	G ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₈ ◁ C₂³×C₈ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

1544 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:

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

Polycyclic presentation:

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

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, 261837);