#6369154

order	512
nilpotency class	3
isoclinism	isoclinic class of #6369150
automorphism	?
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂⁸

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	C₂×D₄²
commutator, Frattini, cosocle	C₄²	C₂⁵
prime radical	#62332	C₂
LCS	G ▷ C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ G
LECS	G ▷ C₄² ▷ C₂² ▷ C₁

1555 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 32
transitive degree	μ_t(G) = 256

A minimal faithful permutation representation — 5 generators on 32 points:

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

Polycyclic presentation:

2^9|a^2=h,b^2=h,c^2=h,f^2=h,g^2=i,b^a=bf,c^a=cg,e^a=ehi,f^a=fh,g^a=gi,c^b=ci,d^b=df,e^b=ei,f^b=fh,e^c=eh,g^c=gi,e^d=ei,f^d=fh

Conjugacy classes & character table

62 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6276350);