#222259

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×AGL(1,ℤ/8)	C₄
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

1094 conjugacy classes of subgroups.

Full subgroup data →

Presentation

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

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

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

Polycyclic presentation:

2^9|a^2=g,d^2=hi,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, 129455);