#459292

order	512
nilpotency class	3
isoclinism	isoclinic class of #453948
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

688 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)(3,6)(4,8)(5,7)(9,10,13,12)(11,15,16,14)(17,18)(19,24)(20,30)(21,26)(22,32)(23,27)(25,31)(28,29);(1,3,5,4)(2,6,7,8)(9,11,13,16)(10,14,12,15)(17,18)(19,28)(20,25)(21,32)(22,26)(23,27)(24,29)(30,31);(2,7)(6,8)(9,12,13,10)(11,15,16,14)(17,19,23,20)(18,24,27,30)(21,31,22,29)(25,32,28,26);(1,4,5,3)(2,8,7,6)(10,12)(14,15)(17,20,23,19)(18,25,27,28)(21,31,22,29)(24,32,30,26)

Polycyclic presentation:

2^9|a^2=e,b^2=ef,c^2=ei,d^2=fi,g^2=i,h^2=i,b^a=be,c^a=cf,d^a=dh,h^a=hi,c^b=cg,g^b=gi,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, 366488);