#383236

order	512
nilpotency class	3
isoclinism	isoclinic class of #30484
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂⁴×C₄²

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₂×C₄	D₄²
commutator	C₂²×C₄	C₂³×C₄
Frattini, cosocle	C₂³×C₄	C₂⁴
socle	C₂²	D₄×C₂²⋊C₄
prime radical	#60366	C₂
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₄ ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

1888 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

80 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 290432);