#6358035

order	512
nilpotency class	3
isoclinism	isoclinic class of #6358021
automorphism	?
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂⁷×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	C₂²×#127
commutator, Frattini, cosocle	C₄²	C₂⁵
prime radical	#62423	C₂
LCS	G ▷ C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₄×#41 ◁ G
LECS	G ▷ C₄² ▷ C₂² ▷ C₁

1906 conjugacy classes of subgroups.

Full subgroup data →

Presentation

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

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

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

Polycyclic presentation:

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

Conjugacy classes & character table

74 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6265231);