#286656

order	512
nilpotency class	3
isoclinism	isoclinic class of #281531
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁴

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

651 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)(17,18,21,26,20,25,22,27)(19,28,30,32,29,23,31,24);(2,5)(6,7)(9,10)(11,13)(12,15)(14,16)(17,19,21,30,20,29,22,31)(18,23,26,24,25,28,27,32);(10,13)(11,16)(14,15)(17,19,20,29)(18,24,25,32)(21,31,22,30)(23,27,28,26);(1,3,4,8)(2,6,5,7)(18,25)(23,28)(24,32)(26,27)

Polycyclic presentation:

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

Conjugacy classes & character table

77 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 193852);