#6352173

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

3142 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:

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

Polycyclic presentation:

2^9|c^2=i,d^2=h,e^2=h,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
maximal normal subgroup of	#72232826

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6259369);