#6365686

order	512
nilpotency class	3
isoclinism	isoclinic class of #6365522
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

1539 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

56 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72306971, #72306977, #72307016
maximal normal subgroup of	#72306971, #72306977, #72307016

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6272882);