#6365329

order	512
nilpotency class	3
isoclinism	isoclinic class of #6365322
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
LCS	G ▷ C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ G
LECS	G ▷ C₄² ▷ C₂² ▷ C₁

1750 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

71 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72338403, #72338419, #72338434
maximal normal subgroup of	#72338403, #72338419, #72338434

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6272525);