#497177

order	512
nilpotency class	2
isoclinism	isoclinic class of #2668
automorphism	?
homology	Abelianization (H₁) = C₂³×C₈ · Schur multiplier (H₂) = C₂²×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, Frattini, cosocle	C₂²×C₈	C₂⁴
commutator, socle, prime radical	C₂³	C₂³×C₈
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂²×C₈ ◁ G
LECS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁

460 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 32
transitive degree	μ_t(G) = 512

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

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

Polycyclic presentation:

2^9|a^2=g,b^2=fg,c^2=ef,d^2=e,g^2=h,h^2=i,b^a=be,c^a=cf,d^a=de,c^b=ce,d^b=di

Conjugacy classes & character table

128 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 404373);