#117680

order	512
nilpotency class	3
isoclinism	isoclinic class of #117671
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	C₂²⋊Q₈
LCS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂²×C₄² ▷ C₂² ▷ C₁

510 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:

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

Polycyclic presentation:

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

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, 24876);