#375870

order	512
nilpotency class	3
isoclinism	isoclinic class of #23903
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂⁷

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₂×C₄	C₂×#120
commutator	C₂⁴	C₂³×C₄
Frattini, cosocle	C₂³×C₄	C₂⁴
socle	C₂²	C₄×#120
prime radical	#23443	C₂
LCS	G ▷ C₂⁴ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₄ ◁ C₂³×C₈ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

1716 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

80 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72262421, #72262422, #72262427, …
maximal normal subgroup of	#72262421, #72262422, #72262427, …
maximal quotient of	#72314867

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 283066);