#6369201

order	512
nilpotency class	3
isoclinism	isoclinic class of #6369150
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

1560 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

62 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72306681, #72306684, #72306686
maximal normal subgroup of	#72306681, #72306684, #72306686

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6276397);