#435544

order	512
nilpotency class	3
isoclinism	isoclinic class of #418769
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁴

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	C₂²×#41	C₂³
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₄ ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

749 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)(5,6,12,16,11,17,10,28)(7,18,20,13,21,33,29,32)(8,14,19,34,23,35,24,31)(9,25,22,30,27,36,26,15);(5,7,12,20,11,21,10,29)(6,13,16,33,17,32,28,18)(8,22,24,9,23,26,19,27)(14,25,31,15,35,36,34,30);(5,8,12,24,11,23,10,19)(6,14,16,31,17,35,28,34)(7,9,20,26,21,27,29,22)(13,30,33,25,32,15,18,36);(2,4)(5,9)(6,15)(7,19)(8,20)(10,22)(11,27)(12,26)(13,31)(14,18)(16,36)(17,30)(21,24)(23,29)(25,28)(32,34)(33,35)

Polycyclic presentation:

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

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, 342740);