#100449

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	C₂²×C₄²	C₂³
prime radical	C₂⁴	#121
LCS	G ▷ C₂×C₄² ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂²×C₄² ▷ C₂³ ▷ C₁

484 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 32
transitive degree	μ_t(G) = 128

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

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

Polycyclic presentation:

2^9|a^2=f,b^2=dfg,c^2=h,d^2=gh,e^2=i,f^2=i,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, 7645);