#117488

order	512
nilpotency class	3
isoclinism	isoclinic class of #117479
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄ · Schur multiplier (H₂) = C₂⁴

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#1439
commutator	C₂×C₄²	C₂²×C₄
Frattini, cosocle	C₂²×C₄²	C₂³
prime radical	C₂⁴	#123
LCS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂²×C₄² ▷ C₂² ▷ C₁

522 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 64
transitive degree	μ_t(G) = 128

A minimal faithful permutation representation — 3 generators on 64 points:

(1,2,6,10,7,12,5,11)(3,13,17,25,16,24,15,8)(4,18,21,9,22,26,20,27)(14,29,31,28,19,23,30,32)(33,34)(35,45)(36,50,54,41)(37,44)(38,42)(39,43)(40,47)(46,61,63,58)(48,56)(49,57)(51,64,62,55)(52,59,53,60);(1,3,5,15,7,16,6,17)(2,8,10,13,12,25,11,24)(4,19,20,31,22,14,21,30)(9,23,26,32,27,29,18,28)(33,35,37,47,39,49,38,48)(34,40,42,45,43,56,44,57)(36,51,52,46,54,62,53,63)(41,58,59,55,50,61,60,64);(1,4,7,22)(2,9,12,27)(3,14,16,19)(5,21,6,20)(8,23,25,29)(10,18,11,26)(13,28,24,32)(15,31,17,30)(33,36,39,54)(34,41,43,50)(35,46,49,63)(37,53,38,52)(40,55,56,64)(42,60,44,59)(45,58,57,61)(47,51,48,62)

Polycyclic presentation:

2^9|a^2=g,b^2=d,c^2=h,d^2=h,e^2=hi,f^2=i,g^2=hi,b^a=bd,c^a=ce,d^a=dh,e^a=ei,f^a=fh,c^b=cf,f^b=fhi,d^c=dh,e^c=ehi,f^c=fi,g^c=gh

Conjugacy classes & character table

62 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 24684);