#6347264

order	512
nilpotency class	3
isoclinism	isoclinic class of #6347252
automorphism	?
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂⁸×C₄

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 ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂² ▷ C₁

3314 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

68 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72213838, #72213906, #72219481, …
maximal normal subgroup of	#72213838, #72213906, #72219481, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6254460);