#189982

order	512
nilpotency class	3
isoclinism	isoclinic class of #163899
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂⁴×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	C₂²×C₂²⋊C₄
commutator	C₂²×C₄	C₂³×C₄
Frattini, cosocle	C₂×C₄²	C₂⁴
prime radical	C₂²×#41	C₂×C₄
LCS	G ▷ C₂²×C₄ ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂×C₄×#41 ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

981 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 40
transitive degree	μ_t(G) = 256

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

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

Polycyclic presentation:

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

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, 97178);