#449584

order	512
nilpotency class	3
isoclinism	isoclinic class of #417544
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁷

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#2404
commutator, Frattini, cosocle	C₂³×C₄	C₂⁴
prime radical	C₂×#585	C₂²
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

2001 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 36
transitive degree	μ_t(G) = 64

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

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

Polycyclic presentation:

2^9|a^2=eh,b^2=i,f^2=i,b^a=be,c^a=cf,d^a=dh,c^b=cg,d^b=de,f^b=fi,e^c=ei,f^c=fi,g^d=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, 356780);