#105776

order	512
nilpotency class	3
isoclinism	isoclinic class of #8465
automorphism	?
homology	Abelianization (H₁) = C₂×C₄² · Schur multiplier (H₂) = C₂⁵

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₂×C₄	#379
commutator	C₂²×C₄	C₂×C₄²
Frattini, cosocle	C₂²×C₄²	C₂³
socle	C₂²	C₂²⋊#95
prime radical	C₂×#121	C₂×C₄
LCS	G ▷ C₂²×C₄ ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂×C₄ ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂²×C₄² ▷ C₂² ▷ C₁

714 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)(3,11,13,7)(4,15,16,8)(5,10)(6,9)(12,19)(14,24,27,20)(17,28,29,21)(18,22)(23,26)(25,30)(31,32)(33,34,38,43,39,44,37,42)(35,45,49,40,48,55,47,54)(36,41,50,56,51,57,52,58)(46,60,61,64,62,59,63,53);(1,3,5,14,6,13,18,27)(2,7,10,20,9,11,22,24)(4,12,17,25,16,26,29,32)(8,19,21,30,15,23,28,31)(33,35)(34,40)(36,46)(37,49)(38,47)(39,48)(41,53)(42,55)(43,45)(44,54)(50,63)(51,62)(52,61)(56,59)(57,64)(58,60);(1,4)(2,8)(3,12)(5,17)(6,16)(7,19)(9,15)(10,21)(11,23)(13,26)(14,25)(18,29)(20,30)(22,28)(24,31)(27,32)(33,36)(34,41)(35,46)(37,52)(38,50)(39,51)(40,53)(42,58)(43,56)(44,57)(45,59)(47,63)(48,62)(49,61)(54,64)(55,60)

Polycyclic presentation:

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

Conjugacy classes & character table

80 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 12972);