#6368150

order	512
nilpotency class	3
isoclinism	isoclinic class of #6368046
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

1659 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)(3,4)(5,13)(6,9)(7,12)(8,11)(10,16)(14,15)(17,18,21,24)(19,22,26,31)(20,23,28,25)(27,29,32,30);(1,3)(2,7)(4,13)(5,15)(6,11)(8,10)(9,14)(12,16)(17,19,21,26)(18,22,24,31)(20,27,28,32)(23,29,25,30);(1,3)(2,8)(4,9)(5,12)(6,11)(7,10)(13,14)(15,16)(18,23)(20,28)(22,29)(24,25)(27,32)(30,31);(1,4,6,8)(2,3,9,11)(5,7,16,14)(10,15,13,12);(2,9)(3,11)(10,13)(12,15)(18,24)(19,26)(23,25)(27,32)

Polycyclic presentation:

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

Conjugacy classes & character table

62 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72339693, #72339713, #72339730
maximal normal subgroup of	#72339693, #72339713, #72339730

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6275346);