#474176

order	512
nilpotency class	3
isoclinism	isoclinic class of #457592
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁴

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#2842
commutator, Frattini, cosocle	C₂³×C₄	C₂⁴
prime radical	C₂³×D₄	C₂³
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

813 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 24
transitive degree	μ_t(G) = 64

A minimal faithful permutation representation — 4 generators on 24 points:

(2,5)(6,8)(9,10,14,16,13,17,12,21)(11,18,22,23,19,24,20,15);(1,2,4,5)(3,6,7,8)(11,19)(20,22);(1,3)(2,6)(4,7)(5,8)(9,11,12,20,13,19,14,22)(10,15,16,18,17,23,21,24);(3,7)(6,8)(11,19)(12,14)(15,23)(16,21)

Polycyclic presentation:

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

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, 381372);