#6346456

order	512
nilpotency class	3
isoclinism	isoclinic class of #18162
automorphism	?
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂⁹

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₂×C₄	C₂×#120
commutator, Frattini, cosocle	C₂²×C₄	C₂⁵
socle	C₂²	C₂²×#120
prime radical	#62773	C₂
LCS	G ▷ C₂²×C₄ ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂×C₄ ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂² ▷ C₁

3200 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:

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

Polycyclic presentation:

2^9|a^2=h,e^2=hi,f^2=h,b^a=bf,c^a=cg,f^a=fh,d^b=dh,f^b=fh,g^b=gi,f^c=fi

Conjugacy classes & character table

80 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72249729, #72280469, #72280498
maximal normal subgroup of	#72249729, #72280469, #72280498
maximal quotient of	#72257870

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6253652);