#118160

order	512
nilpotency class	3
isoclinism	isoclinic class of #10645
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄ · Schur multiplier (H₂) = C₂⁴

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₂×C₄	#397
commutator	C₂×C₄²	C₂²×C₄
Frattini, cosocle	C₂²×C₄²	C₂³
socle	C₂²	#1439
prime radical	#3444	C₂²
LCS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂×C₄ ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂²×C₄² ▷ C₂² ▷ C₁

527 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:

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

Polycyclic presentation:

2^9|a^2=g,b^2=d,d^2=h,e^2=h,f^2=i,g^2=hi,b^a=bd,c^a=ce,d^a=dh,e^a=ehi,c^b=cf,e^b=ei,d^c=di,e^c=eh,f^c=fi,g^c=gi

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, 25356);