#383882

order	512
nilpotency class	3
isoclinism	isoclinic class of #31077
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

1525 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)(8,23)(9,15)(10,14)(11,16)(12,20)(13,25)(18,33)(19,22)(21,29)(24,34)(26,31)(27,32)(28,36)(30,35);(5,7,10,21,11,22,9,20)(6,12,15,19,16,29,14,17)(8,24,25,18,27,35,26,36)(13,30,23,33,31,34,32,28);(5,8)(6,13)(7,18)(9,25)(10,26)(11,27)(12,28)(14,23)(15,32)(16,31)(17,30)(19,34)(20,35)(21,24)(22,36)(29,33);(5,6)(7,19)(8,13)(9,15)(10,14)(11,16)(12,21)(17,22)(18,34)(20,29)(23,26)(24,28)(25,32)(27,31)(30,36)(33,35)

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=de,f^c=fi,g^c=gi,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, 291078);