#383883

order	512
nilpotency class	3
isoclinism	isoclinic class of #31076
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂³×C₄²

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₂×C₄	H(3,𝔽₄)
commutator	C₂²×C₄	C₂³×C₄
Frattini, cosocle	C₂³×C₄	C₂⁴
socle	C₂²	#2684
prime radical	C₂×#585	C₂²
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₄ ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

1535 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:

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

Polycyclic presentation:

2^9|a^2=h,b^2=f,e^2=i,f^2=i,g^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,g^c=gi,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, 291079);