#6357029

order	512
nilpotency class	3
isoclinism	isoclinic class of #6356901
automorphism	?
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂⁸

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	C₂²×#121
commutator, Frattini, cosocle	C₄²	C₂⁵
LCS	G ▷ C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₄×#41 ◁ G
LECS	G ▷ C₄² ▷ C₂² ▷ C₁

1649 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

65 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72338351, #72338426, #72338448
maximal normal subgroup of	#72338351, #72338426, #72338448

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6264225);