#6508694

order	512
nilpotency class	3
isoclinism	isoclinic class of #6499870
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	#62817	C₂
LCS	G ▷ C₂⁴ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂⁶ ◁ G
LECS	G ▷ C₂⁴ ▷ C₂ ▷ C₁

3826 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

68 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72238047, #72251183, #72251186, …
maximal normal subgroup of	#72238047, #72251183, #72251186, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6415890);