#444610

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	C₂⁵	C₂×D₄
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂⁴×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

1206 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:

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

Polycyclic presentation:

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

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, 351806);