#466100

order	512
nilpotency class	3
isoclinism	isoclinic class of #454426
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂²×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	D₄²
commutator, Frattini, cosocle	C₂³×C₄	C₂⁴
prime radical	C₂⁴	H(5,𝔽₂)
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

769 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:

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

Polycyclic presentation:

2^9|a^2=e,b^2=fg,c^2=ei,d^2=f,g^2=i,h^2=i,b^a=be,c^a=cf,d^a=dh,h^a=hi,c^b=cg,d^c=de,g^c=gi,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, 373296);