#498776

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

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

414 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:

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

Polycyclic presentation:

2^9|a^2=g,b^2=f,c^2=f,d^2=e,g^2=h,h^2=i,b^a=be,c^a=cf,d^a=def,c^b=cefi,d^b=de

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, 405972);