#121819

order	512
nilpotency class	3
isoclinism	isoclinic class of #10645
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄ · Schur multiplier (H₂) = C₂³

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	#1445
prime radical	C₂⁴	C₂²⋊Q₈
LCS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂×C₄ ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂²×C₄² ▷ C₂² ▷ C₁

368 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:

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

Polycyclic presentation:

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

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, 29015);