#117498

order	512
nilpotency class	3
isoclinism	isoclinic class of #117489
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

514 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)(3,13,16,24,17,28,15,8)(4,18,21,9,22,26,20,27)(14,23,31,32,19,25,30,29)(33,34)(35,44)(36,49,53,41)(37,54)(38,42)(39,43)(40,46)(45,60,63,57)(47,61)(48,56)(50,55,62,64)(51,58,52,59);(1,3,5,15,7,17,6,16)(2,8,10,13,12,24,11,28)(4,19,20,31,22,14,21,30)(9,25,26,29,27,23,18,32)(33,35,37,46,39,47,38,48)(34,40,42,44,43,56,54,61)(36,50,51,45,53,62,52,63)(41,57,58,64,49,60,59,55);(1,4,7,22)(2,9,12,27)(3,14,17,19)(5,21,6,20)(8,23,24,25)(10,18,11,26)(13,29,28,32)(15,31,16,30)(33,36,39,53)(34,41,43,49)(35,45,47,63)(37,52,38,51)(40,55,56,64)(42,59,54,58)(44,60,61,57)(46,50,48,62)

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,c^b=cf,e^b=eh,f^b=fhi,d^c=dh,e^c=ehi,f^c=fi,g^c=gh

Conjugacy classes & character table

56 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 24694);