#121001

order	512
nilpotency class	3
isoclinism	isoclinic class of #120998
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄ · Schur multiplier (H₂) = C₂³

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#1442
commutator	C₂×C₄²	C₂²×C₄
Frattini, cosocle	C₂²×C₄²	C₂³
prime radical	C₂⁴	#123
LCS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂²×C₄² ▷ C₂² ▷ C₁

441 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:

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

Polycyclic presentation:

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

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, 28197);