#461540

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	C₂²×AGL(1,ℤ/8)	C₂²
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

1017 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:

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

Polycyclic presentation:

2^9|a^2=eg,b^2=ei,c^2=ef,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, 368736);