#6342500

order	512
nilpotency class	2
isoclinism	isoclinic class of #3532
automorphism	Aut(G) order 131072
homology	Abelianization (H₁) = C₂³×C₄² · Schur multiplier (H₂) = C₂⁷×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, Frattini, cosocle	C₄²	C₂⁵
commutator, socle	C₂²	C₂³×C₄²
prime radical	#3444	C₂²
LCS	G ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₄² ◁ G
LECS	G ▷ C₄² ▷ C₂² ▷ C₁

1917 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:

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

Polycyclic presentation:

2^9|a^2=f,b^2=g,f^2=h,g^2=i,b^a=bi,d^a=dh,e^a=eh,d^c=di,e^c=eh

Conjugacy classes & character table

152 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72286982, #72287061, #72287154, …
maximal normal subgroup of	#72286982, #72287061, #72287154, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6249696);