#121380

order	512
nilpotency class	3
isoclinism	isoclinic class of #10218
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	C₄⋊Q₈
LCS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂×C₄ ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂²×C₄² ▷ C₂² ▷ C₁

542 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

68 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 28576);