#323010

order	512
nilpotency class	3
isoclinism	isoclinic class of #289819
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂³×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	D₄²
commutator, Frattini, cosocle	C₂×C₄²	C₂⁴
prime radical	#2541	C₂²
LCS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂×C₄² ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

768 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 36
transitive degree	μ_t(G) = 256

A minimal faithful permutation representation — 4 generators on 36 points:

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

Polycyclic presentation:

2^9|a^2=f,b^2=eg,c^2=h,d^2=eh,e^2=h,f^2=i,g^2=i,b^a=be,c^a=cf,d^a=de,e^a=eh,c^b=cg,d^b=dhi,f^c=fi,g^c=gi

Conjugacy classes & character table

62 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 230206);