#6356930

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

1618 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)(3,10)(4,8)(5,15)(6,9)(7,14)(11,16)(12,13)(18,22)(19,26)(20,23)(25,31)(28,29)(30,32);(1,3,4,13,6,11,5,14)(2,7,15,16,9,12,8,10)(17,18,23,31,21,25,20,22)(19,27,26,30,29,24,28,32);(3,11)(7,12)(10,16)(13,14)(17,19,21,29)(18,24,25,27)(20,28,23,26)(22,30,31,32);(1,2)(3,12)(4,15)(5,8)(6,9)(7,11)(10,13)(14,16)(18,25)(22,31)(24,27)(30,32);(1,4,6,5)(2,8,9,15)(3,13,11,14)(7,10,12,16)

Polycyclic presentation:

2^9|b^2=fh,c^2=i,e^2=hi,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	#72339869, #72340119, #72340121, …
maximal normal subgroup of	#72339869, #72340119, #72340121, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6264126);