#121196

order	512
nilpotency class	3
isoclinism	isoclinic class of #10631
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄ · Schur multiplier (H₂) = C₂⁴

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	#1443
prime radical	C₂⁴	#121
LCS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂×C₄ ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂²×C₄² ▷ C₂² ▷ C₁

476 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 40
transitive degree	μ_t(G) = 64

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

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

Polycyclic presentation:

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

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, 28392);