#6477581

order	512
nilpotency class	3
isoclinism	isoclinic class of #6470225
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	C₂×#3490	C₂
LCS	G ▷ C₂⁴ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂⁶ ◁ G
LECS	G ▷ C₂⁴ ▷ C₂ ▷ C₁

3814 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

74 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72238044, #72238047, #72238050, …
maximal normal subgroup of	#72238044, #72238047, #72238050, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6384777);