#441198

order	512
nilpotency class	3
isoclinism	isoclinic class of #417464
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁴×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	C₂×#120
commutator, Frattini, cosocle	C₂³×C₄	C₂⁴
prime radical	C₂×#535	C₂²
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

1403 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:

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

Polycyclic presentation:

2^9|a^2=efg,b^2=eg,c^2=e,d^2=ei,f^2=i,h^2=i,b^a=be,c^a=cf,d^a=dh,h^a=hi,c^b=cg,d^b=dei,f^c=fi,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, 348394);