#223004

order	512
nilpotency class	3
isoclinism	isoclinic class of #24269
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂⁴

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₂²×C₄	C₂²×D₄
commutator	C₂²×C₄	C₂³×C₄
Frattini, cosocle	C₂×C₄²	C₂⁴
socle	C₂³	C₂²×C₂²⋊C₄
prime radical	#2535	C₄
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂²×C₄ ◁ C₄²×D₄ ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

886 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:

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

Polycyclic presentation:

2^9|a^2=g,c^2=eh,f^2=h,g^2=i,b^a=be,c^a=cf,d^a=dhi,f^a=fh,c^b=chi,d^b=dei,d^c=di,f^c=fh

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, 130200);