#6496274

order	512
nilpotency class	3
isoclinism	isoclinic class of #6470231
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	C₂⁴	C₂⁵
prime radical	#62846	C₂
LCS	G ▷ C₂⁴ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂²×C₂²⋊C₄ ◁ G
LECS	G ▷ C₂⁴ ▷ C₂ ▷ C₁

3154 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 32
transitive degree	μ_t(G) = 64

A minimal faithful permutation representation — 5 generators on 32 points:

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

Polycyclic presentation:

2^9|a^2=hi,e^2=h,b^a=bf,c^a=cg,e^a=eh,c^b=ch,d^b=dh,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	#72316108
maximal normal subgroup of	#72316108

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6403470);