#115836

order	512
nilpotency class	3
isoclinism	isoclinic class of #10046
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄ · Schur multiplier (H₂) = C₂⁴

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₂²×C₄	#120
commutator	C₂×C₄²	C₂²×C₄
Frattini, cosocle	C₂²×C₄²	C₂³
socle	C₂³	#386
prime radical	C₂⁴	#123
LCS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂²×C₄ ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂²×C₄² ▷ C₂² ▷ C₁

702 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 28
transitive degree	μ_t(G) = 256

A minimal faithful permutation representation — 3 generators on 28 points:

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

Polycyclic presentation:

2^9|a^2=g,b^2=d,c^2=h,e^2=h,f^2=i,g^2=i,b^a=bd,c^a=ce,e^a=eh,f^a=fi,c^b=cf,e^b=ei,f^b=fi,e^c=eh,f^c=fi

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, 23032);