#478445

order	512
nilpotency class	3
isoclinism	isoclinic class of #476095
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂×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	#2782
prime radical	C₂×#575	C₂²
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₄ ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

792 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)(5,6,11,17,12,18,10,16)(7,19,30,13,21,35,20,32)(8,22,24,31,26,34,25,14)(9,27,23,33,29,15,28,36);(5,7,11,20,12,21,10,30)(6,13,17,19,18,32,16,35)(8,23,24,9,26,28,25,29)(14,33,22,27,31,36,34,15);(5,8,10,24,12,26,11,25)(6,14,16,22,18,31,17,34)(7,9,20,23,21,29,30,28)(13,27,19,33,32,15,35,36);(2,4)(5,9)(6,15)(7,8)(10,28)(11,23)(12,29)(13,31)(14,32)(16,33)(17,36)(18,27)(19,22)(20,25)(21,26)(24,30)(34,35)

Polycyclic presentation:

2^9|a^2=efg,b^2=f,c^2=e,e^2=i,f^2=i,g^2=i,b^a=be,c^a=cf,d^a=dh,c^b=cg,d^b=def,e^b=ei,h^b=hi,d^c=de,f^c=fi,h^c=hi,e^d=ei,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, 385641);