#6352180

order	512
nilpotency class	3
isoclinism	isoclinic class of #6352157
automorphism	?
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂⁹

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 ◁ C₂×C₄×D₄ ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂² ▷ C₁

3167 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

68 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72232826, #72234735, #72271417, …
maximal normal subgroup of	#72232826, #72234735, #72338739
maximal quotient of	#72268213

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6259376);