#220199

order	512
nilpotency class	3
isoclinism	isoclinic class of #220190
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×#41	C₂×C₄
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

1020 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:

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

Polycyclic presentation:

2^9|a^2=g,b^2=hi,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=dei,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, 127395);