#383235

order	512
nilpotency class	3
isoclinism	isoclinic class of #30490
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

1878 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:

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

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=de,f^c=fi,e^d=ei,f^d=fi

Conjugacy classes & character table

68 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 290431);