#6347406

order	512
nilpotency class	3
isoclinism	isoclinic class of #6347398
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

3144 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)(8,21)(9,16)(10,19)(12,14)(13,28)(17,30)(20,25)(22,32)(23,29)(24,31)(26,27);(1,2)(3,8)(4,16)(5,9)(6,10)(7,15)(11,25)(12,22)(13,29)(14,23)(17,31)(18,24)(19,30)(20,27)(21,28)(26,32);(3,12)(4,17)(7,19)(8,22)(11,14)(13,26)(15,30)(16,31)(20,21)(23,25)(27,28)(29,32);(1,3,6,14)(2,8,10,23)(4,13,17,28)(5,12,18,11)(7,20,19,32)(9,22,24,25)(15,27,30,26)(16,29,31,21);(2,9)(7,16)(8,22)(10,24)(19,31)(20,29)(21,32)(23,25)

Polycyclic presentation:

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

Conjugacy classes & character table

62 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72232996, #72234669, #72234685
maximal normal subgroup of	#72232996, #72234669, #72234685

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6254602);