#445101

order	512
nilpotency class	3
isoclinism	isoclinic class of #417901
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	C₂²×#41	C₂³
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₄ ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

703 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)(7,17,21,12)(8,22)(9,14)(10,16)(11,15)(13,24)(18,30,35,36)(19,28,20,29)(23,33,34,27)(25,32)(26,31);(5,7,9,19,11,21,10,20)(6,12,14,29,15,17,16,28)(8,23,24,35,26,34,25,18)(13,30,31,33,32,36,22,27);(5,8,10,25,11,26,9,24)(6,13,14,31,15,32,16,22)(7,18,19,23,21,35,20,34)(12,27,28,36,17,33,29,30);(2,4)(6,14,15,16)(12,28,17,29)(13,31,32,22)(27,36,33,30)

Polycyclic presentation:

2^9|a^2=efg,b^2=ef,c^2=eg,d^2=e,f^2=i,g^2=i,h^2=i,b^a=be,c^a=cf,d^a=dh,e^a=ei,f^a=fi,h^a=hi,c^b=cg,d^b=de,g^b=gi,h^b=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, 352297);