#444434

order	512
nilpotency class	3
isoclinism	isoclinic class of #418500
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁴

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

826 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:

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

Polycyclic presentation:

2^9|a^2=efi,b^2=ef,c^2=g,d^2=ef,g^2=i,h^2=i,b^a=be,c^a=cf,d^a=dh,h^a=hi,c^b=cg,d^b=de,g^b=gi,h^d=hi

Conjugacy classes & character table

68 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 351630);