#243759

order	512
nilpotency class	3
isoclinism	isoclinic class of #233100
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

638 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)(3,6)(4,5)(7,8)(9,10,14,15,13,11,12,16)(17,18,21,25,20,24,30,23)(19,26,27,32,28,22,29,31);(2,5)(6,8)(17,19,20,28)(18,22,24,26)(21,27,30,29)(23,32,25,31);(9,11,13,10)(12,16,14,15)(18,23)(19,27)(21,30)(24,25)(28,29)(31,32);(1,3,4,7)(2,6,5,8)(9,12,13,14)(10,15,11,16)

Polycyclic presentation:

2^9|a^2=fhi,b^2=i,c^2=hi,d^2=eh,f^2=h,g^2=i,b^a=be,c^a=cf,d^a=dhi,c^b=cg,g^b=gi,f^c=fh,g^c=gi

Conjugacy classes & character table

68 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 150955);