#469728

order	512
nilpotency class	3
isoclinism	isoclinic class of #453996
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂³×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	C₂×#121
commutator, Frattini, cosocle	C₂³×C₄	C₂⁴
prime radical	C₂⁵	#41
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂⁴×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

1002 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

62 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 376924);