#462573

order	512
nilpotency class	3
isoclinism	isoclinic class of #454426
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂×C₄²

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	D₄²
commutator, Frattini, cosocle	C₂³×C₄	C₂⁴
prime radical	C₂×#537	C₂²
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

943 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 32
transitive degree	μ_t(G) = 128

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

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

Polycyclic presentation:

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

Conjugacy classes & character table

62 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 369769);