#110527

order	512
nilpotency class	3
isoclinism	isoclinic class of #9345
automorphism	?
homology	Abelianization (H₁) = C₂×C₄² · Schur multiplier (H₂) = C₂³×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₂×C₄	#386
commutator	C₂²×C₄	C₂×C₄²
Frattini, cosocle	C₂²×C₄²	C₂³
socle	C₂²	#1437
prime radical	C₂⁴	C₄×D₄
LCS	G ▷ C₂²×C₄ ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂×C₄ ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂²×C₄² ▷ C₂² ▷ C₁

602 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

80 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 17723);