#497263

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	C₂³	C₂³×C₈
prime radical	C₂²×D₄	C₂×C₈
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂²×C₈ ◁ G
LECS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁

525 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 32
transitive degree	μ_t(G) = 256

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

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

Polycyclic presentation:

2^9|a^2=g,b^2=eg,c^2=e,d^2=ef,g^2=h,h^2=i,b^a=be,c^a=cf,d^a=de,c^b=cei,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, 404459);