#100354

order	512
nilpotency class	3
isoclinism	isoclinic class of #100251
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄ · Schur multiplier (H₂) = C₂²×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	#386
commutator	C₂×C₄²	C₂²×C₄
Frattini, cosocle	C₄³	C₂³
prime radical	C₂²×D₄	C₂²⋊C₄
LCS	G ▷ C₂×C₄² ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₄³ ◁ G
LECS	G ▷ C₄³ ▷ C₂³ ▷ C₁

477 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 32
transitive degree	μ_t(G) = 256

A minimal faithful permutation representation — 3 generators on 32 points:

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

Polycyclic presentation:

2^9|a^2=f,b^2=dfgi,c^2=hi,d^2=gh,e^2=i,f^2=h,b^a=bd,c^a=ce,d^a=dg,c^b=cg,d^b=dh,e^b=ei,f^b=fh,d^c=di,e^c=ei,f^c=fi

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, 7550);