#345123

order	512
nilpotency class	3
isoclinism	isoclinic class of #334823
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂³

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	D₄²
commutator, Frattini, cosocle	C₂×C₄²	C₂⁴
prime radical	C₂²×D₄	C₂×D₄
LCS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂×C₄² ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

603 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 32
transitive degree	μ_t(G) = 256

A minimal faithful permutation representation — 4 generators on 32 points:

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

Polycyclic presentation:

2^9|a^2=ef,b^2=ehi,c^2=eh,d^2=h,f^2=h,g^2=i,b^a=be,c^a=cf,d^a=defh,c^b=cg,d^b=de,g^b=gi,f^c=fh,g^c=gi,f^d=fh

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, 252319);