#477177

order	512
nilpotency class	3
isoclinism	isoclinic class of #476095
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂³×C₄²

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₂×C₄	H(3,𝔽₄)
commutator, Frattini, cosocle	C₂³×C₄	C₂⁴
socle	C₂²	#2684
prime radical	#23406	C₂
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₄ ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

1111 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

56 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 384373);