#472431

order	512
nilpotency class	3
isoclinism	isoclinic class of #455110
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂³

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	C₂×#121
commutator, Frattini, cosocle	C₂³×C₄	C₂⁴
prime radical	C₂⁴	C₂×#41
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

706 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)(3,5)(4,6)(7,8)(9,10,12,13)(11,15,16,14)(17,18,22,26)(19,27,29,24)(20,21,25,23)(28,32,31,30);(1,3,4,7)(2,5,6,8)(10,13)(14,15)(17,19,21,31,22,29,23,28)(18,24,25,32,26,27,20,30);(11,16)(14,15)(17,20,23,18,22,25,21,26)(19,24,28,32,29,27,31,30);(2,6)(5,8)(9,11,12,16)(10,14,13,15)(17,21,22,23)(18,25,26,20)(19,28,29,31)(24,30,27,32)

Polycyclic presentation:

2^9|a^2=e,b^2=egh,c^2=f,d^2=e,f^2=i,g^2=i,b^a=be,c^a=cf,d^a=dh,f^a=fi,c^b=cg,d^b=di,d^c=de,g^c=gi

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, 379627);