#286974

order	512
nilpotency class	3
isoclinism	isoclinic class of #281787
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	C₂×C₄²	C₂⁴
prime radical	C₂×#127	C₂³
LCS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₄³ ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

602 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)(3,7)(4,5)(6,8)(9,10,14,15,12,11,13,16)(17,18,22,24,21,26,20,25)(19,23,29,32,27,31,28,30);(2,5)(6,7)(9,11,13,15,12,10,14,16)(17,19,20,28,21,27,22,29)(18,23,25,30,26,31,24,32);(10,15)(11,16)(13,14)(18,24)(19,27)(20,22)(23,30)(25,26)(31,32);(1,3,4,8)(2,6,5,7)(10,11)(15,16)(19,27)(23,31)(28,29)(30,32)

Polycyclic presentation:

2^9|a^2=fh,b^2=f,d^2=ehi,f^2=h,g^2=hi,b^a=be,c^a=cf,d^a=de,c^b=cg,d^b=dh,g^b=gi,f^c=fh,g^c=ghi

Conjugacy classes & character table

65 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 194170);