#181254

order	512
nilpotency class	3
isoclinism	isoclinic class of #181131
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄² · Schur multiplier (H₂) = C₂⁵×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	C₂²×#95
commutator	C₂×C₄	C₂²×C₄²
Frattini, cosocle	C₂³×C₄	C₂⁴
prime radical	C₂⁴	C₂²×D₄
LCS	G ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂²×C₄×Q₈ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂² ▷ C₁

1461 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:

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

Polycyclic presentation:

2^9|a^2=f,b^2=g,c^2=h,d^2=i,e^2=hi,g^2=hi,b^a=be,e^a=eh,g^a=ghi,c^b=ch,f^b=fi,d^c=di

Conjugacy classes & character table

92 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 88450);