#6462906

order	512
nilpotency class	3
isoclinism	isoclinic class of #6442597
automorphism	?
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂⁹

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₄	C₂×#535
commutator, Frattini, cosocle	C₂²×C₄	C₂⁵
socle	C₂	C₂×#2535
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₄ ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

2342 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)(3,12)(4,9)(5,22)(6,10)(7,11)(8,15)(13,16)(14,25)(17,24)(18,31)(19,26)(20,32)(21,27)(23,30)(28,29);(1,3)(2,8)(4,18)(5,17)(6,15)(7,16)(9,25)(10,13)(11,24)(12,22)(14,21)(19,30)(20,29)(23,27)(26,28)(31,32);(1,2)(3,13,16,12)(4,9)(5,22)(6,10)(7,11)(8,15,24,17)(14,28,29,25)(18,23,30,31)(19,26)(20,32)(21,27);(1,4,6,21,7,19,5,20)(2,9,10,27,11,26,22,32)(3,14,17,18,16,29,15,30)(8,23,12,25,24,31,13,28);(4,19)(9,26)(14,29)(18,30)(20,21)(23,31)(25,28)(27,32)

Polycyclic presentation:

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

Conjugacy classes & character table

80 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72216029, #72216037, #72216095, …
maximal normal subgroup of	#72216029, #72216037, #72216095, …
maximal quotient of	C₂×#6462906, #72255634, #72256293, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6370102);