#6347375

order	512
nilpotency class	3
isoclinism	isoclinic class of #6347363
automorphism	?
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂⁸×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	C₂²×#120
commutator, Frattini, cosocle	C₂²×C₄	C₂⁵
LCS	G ▷ C₂²×C₄ ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂² ▷ C₁

2964 conjugacy classes of subgroups.

Full subgroup data →

Presentation

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

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

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

Polycyclic presentation:

2^9|d^2=hi,f^2=h,b^a=bf,c^a=cg,d^a=di,f^a=fh,e^b=eh,f^b=fh,g^b=gi,d^c=dh,f^c=fi

Conjugacy classes & character table

68 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6254571);