#445048

order	512
nilpotency class	3
isoclinism	isoclinic class of #420064
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂²×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₂×C₄	#555
commutator, Frattini, cosocle	C₂³×C₄	C₂⁴
socle	C₂²	#2495
prime radical	C₂⁴	2₋¹⁺⁴
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₄ ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

592 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)(5,6,12,28,11,15,10,9)(7,16,20,13,21,27,19,18)(8,22,24,14,26,23,25,30)(17,32,33,36,34,29,35,31);(5,7,10,20,11,21,12,19)(6,13,9,27,15,18,28,16)(8,17,25,33,26,34,24,35)(14,29,22,31,30,32,23,36);(5,8,12,25,11,26,10,24)(6,14,28,22,15,30,9,23)(7,17,19,33,21,34,20,35)(13,29,16,31,18,32,27,36);(2,4)(5,9,10,15,11,28,12,6)(7,18,19,27,21,13,20,16)(8,23,24,30,26,22,25,14)(17,32,33,36,34,29,35,31)

Polycyclic presentation:

2^9|a^2=fg,b^2=f,c^2=eg,d^2=e,e^2=i,f^2=i,b^a=be,c^a=cf,d^a=dh,c^b=cg,d^b=de,e^b=ei,g^b=gi,f^c=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, 352244);