#299941

order	512
nilpotency class	3
isoclinism	isoclinic class of #299869
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁴×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#2433
commutator, Frattini, cosocle	C₂×C₄²	C₂⁴
LCS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

996 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

74 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 207137);