#119740

order	512
nilpotency class	3
isoclinism	isoclinic class of #119739
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄ · Schur multiplier (H₂) = C₂⁴

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#1440
commutator	C₂×C₄²	C₂²×C₄
Frattini, cosocle	C₂²×C₄²	C₂³
prime radical	C₂⁴	#121
LCS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂²×C₄² ▷ C₂² ▷ C₁

659 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)(3,5)(4,7)(6,8)(9,10)(11,21)(12,26,30,35)(13,19)(14,18)(15,20)(16,23)(17,28,36,29)(22,31,39,40)(24,33)(25,32)(27,37,38,34);(3,8)(5,6)(9,11,15,25)(10,16,20,33)(12,27,30,38)(13,24,14,23)(17,34,36,37)(18,32,19,21)(22,28,39,29)(26,31,35,40);(2,5)(3,8)(6,7)(9,12,13,28,15,30,14,29)(10,17,18,35,20,36,19,26)(11,22,23,38,25,39,24,27)(16,31,21,37,33,40,32,34)

Polycyclic presentation:

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

Conjugacy classes & character table

56 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 26936);