#243830

order	512
nilpotency class	3
isoclinism	isoclinic class of #10012
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁴

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₄²	#127
commutator, Frattini, cosocle	C₂×C₄²	C₂⁴
socle	C₂²	C₂×#399
prime radical	C₂²×#41	Q₈
LCS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₄² ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

726 conjugacy classes of subgroups.

Full subgroup data →

Presentation

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

A minimal faithful permutation representation — 4 generators on 32 points:

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

Polycyclic presentation:

2^9|a^2=e,b^2=ef,d^2=h,e^2=h,f^2=i,b^a=be,c^a=cf,g^a=ghi,c^b=cg,e^b=eh,f^b=fh,e^c=ei,f^c=fi

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, 151026);