#121496

order	512
nilpotency class	3
isoclinism	isoclinic class of #9996
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄ · Schur multiplier (H₂) = C₂⁴

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₂×C₄	#392
commutator	C₂×C₄²	C₂²×C₄
Frattini, cosocle	C₂²×C₄²	C₂³
socle	C₂²	#1444
prime radical	C₂⁴	C₂²⋊Q₈
LCS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂×C₄ ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂²×C₄² ▷ C₂² ▷ C₁

559 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 40
transitive degree	μ_t(G) = 128

A minimal faithful permutation representation — 3 generators on 40 points:

(2,6)(3,4)(7,8)(9,10,14,18,15,19,13,29)(11,20,24,31,23,35,22,16)(12,17,28,32,27,33,26,39)(21,36,38,30,25,34,37,40);(1,2,3,6,5,8,4,7)(9,11,13,22,15,23,14,24)(10,16,18,20,19,31,29,35)(12,25,26,38,27,21,28,37)(17,30,32,34,33,40,39,36);(2,6)(3,4)(7,8)(9,12)(10,17)(11,21,23,25)(13,26)(14,28)(15,27)(16,30,31,40)(18,32)(19,33)(20,34,35,36)(22,37,24,38)(29,39)

Polycyclic presentation:

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

Conjugacy classes & character table

74 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 28692);