#431966

order	512
nilpotency class	3
isoclinism	isoclinic class of #417901
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁴×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	#23399	C₂
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₄ ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

1188 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)(7,13)(8,22,25,24)(9,10)(11,15)(12,16)(14,26,32,27)(17,31,34,33)(18,19)(20,29)(21,30)(23,28,35,36);(1,2)(3,4)(5,7,10,19,12,21,11,20)(6,13,9,18,16,30,15,29)(8,23,26,17,25,35,27,34)(14,31,22,36,32,33,24,28);(2,4)(5,8,10,27,12,25,11,26)(6,14,9,24,16,32,15,22)(7,17,19,23,21,34,20,35)(13,28,18,33,30,36,29,31);(5,9,10,16,12,15,11,6)(7,18,19,30,21,29,20,13)(8,24,27,32,25,22,26,14)(17,33,23,36,34,31,35,28)

Polycyclic presentation:

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

Conjugacy classes & character table

74 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 339162);