#359034

order	512
nilpotency class	3
isoclinism	isoclinic class of #2002
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂⁷

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₂×C₈	#120
commutator	C₂²×C₄	C₂³×C₄
Frattini, cosocle	C₂³×C₄	C₂⁴
socle	C₂²	C₄×#120
prime radical	#23406	C₂
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₈ ◁ C₂³×C₈ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

1864 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

104 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 266230);