#420762

order	512
nilpotency class	3
isoclinism	isoclinic class of #417902
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁵×C₄

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	C₂×#3018	C₂
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₄ ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

1867 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

62 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 327958);