#222269

order	512
nilpotency class	3
isoclinism	isoclinic class of #222256
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂⁵

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	D₄×C₂²⋊C₄
commutator	C₂²×C₄	C₂³×C₄
Frattini, cosocle	C₂×C₄²	C₂⁴
prime radical	C₂²×#135	C₄
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

1049 conjugacy classes of subgroups.

Full subgroup data →

Presentation

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

A minimal faithful permutation representation — 4 generators on 40 points:

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

Polycyclic presentation:

2^9|a^2=g,b^2=h,c^2=h,d^2=h,e^2=h,g^2=i,b^a=be,c^a=cf,d^a=di,e^a=eh,f^a=fh,d^b=de,e^b=eh,g^c=gh,e^d=eh

Conjugacy classes & character table

80 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 129465);