#111774

order	512
nilpotency class	3
isoclinism	isoclinic class of #10260
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄ · Schur multiplier (H₂) = C₂⁵

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle, prime radical	C₂⁴	#121
commutator	C₂×C₄²	C₂²×C₄
Frattini, cosocle	C₂²×C₄²	C₂³
LCS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂⁴ ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂²×C₄² ▷ C₂² ▷ C₁

1054 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 32
transitive degree	μ_t(G) = 512

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

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

Polycyclic presentation:

2^9|a^2=g,b^2=i,c^2=h,d^2=i,f^2=h,g^2=hi,b^a=bd,c^a=ce,d^a=dh,f^a=fh,c^b=cf,d^b=di,f^b=fh,g^b=ghi,d^c=dh,f^c=fh

Conjugacy classes & character table

92 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 18970);