#243191

order	512
nilpotency class	3
isoclinism	isoclinic class of #233152
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂²×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	C₂×#121
commutator, Frattini, cosocle	C₂×C₄²	C₂⁴
prime radical	C₂⁴	#121
LCS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

596 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)(3,8)(4,6)(5,7)(9,10,13,17,12,16,21,15)(11,18,20,23,19,14,22,24);(1,3,5,4)(2,6,7,8)(9,11,12,19)(10,14,16,18)(13,20,21,22)(15,23,17,24)(25,26)(27,34)(28,32)(29,37)(30,33)(31,35)(36,39)(38,40);(1,4,5,3)(2,6,7,8)(10,15)(13,21)(14,23)(16,17)(18,24)(20,22)(25,27)(26,31)(28,35)(29,38)(30,36)(32,39)(33,40)(34,37);(11,19)(14,18)(20,22)(23,24)(25,28,30,29)(26,32,33,37)(27,35,36,38)(31,39,40,34)

Polycyclic presentation:

2^9|a^2=f,b^2=e,c^2=eh,d^2=i,f^2=h,g^2=i,b^a=be,c^a=cf,c^b=cg,d^b=dh,g^b=gi,f^c=fh,g^c=gi

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, 150387);