#245804

order	512
nilpotency class	3
isoclinism	isoclinic class of #245711
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁶

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#2403
commutator, Frattini, cosocle	C₂×C₄²	C₂⁴
LCS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

1207 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 32
transitive degree	μ_t(G) = 64

A minimal faithful permutation representation — 4 generators on 32 points:

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

Polycyclic presentation:

2^9|b^2=i,c^2=hi,d^2=hi,e^2=h,f^2=i,b^a=be,c^a=cf,d^a=de,e^a=eh,f^a=fi,g^a=gi,c^b=cg,e^b=eh,e^c=ei,f^c=fi,e^d=eh,f^d=fi

Conjugacy classes & character table

56 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 153000);