#6348994

order	512
nilpotency class	3
isoclinism	isoclinic class of #6348967
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	C₄²	C₂⁵
LCS	G ▷ C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₄² ▷ C₂² ▷ C₁

3057 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)(3,11)(4,8)(5,9)(6,10)(7,12)(13,15)(14,16)(17,18)(19,28)(20,27)(21,26)(22,25)(23,30)(24,29)(31,32);(1,3)(2,7)(4,12)(5,14)(6,13)(8,15)(9,11)(10,16)(17,19)(18,23)(20,29)(21,30)(22,31)(24,25)(26,32)(27,28);(17,19)(18,24)(20,29)(21,30)(22,31)(23,25)(26,28)(27,32);(1,4,6,5)(2,8,10,9)(3,12,13,14)(7,15,16,11)(18,25)(23,24)(26,27)(28,32);(1,4,6,5)(2,9,10,8)(3,12,13,14)(7,11,16,15)(17,20,22,21)(18,26,25,27)(19,29,31,30)(23,32,24,28)

Polycyclic presentation:

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

Conjugacy classes & character table

74 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72231252, #72231256, #72231264, …
maximal normal subgroup of	#72231252, #72231256, #72231264, …
maximal quotient of	#72279227, #72279282, C₂×#6348994

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6256190);