#478168

order	512
nilpotency class	3
isoclinism	isoclinic class of #476092
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

790 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:

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

Polycyclic presentation:

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

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, 385364);