#459712

order	512
nilpotency class	3
isoclinism	isoclinic class of #459488
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂³×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#2684
commutator, Frattini, cosocle	C₂³×C₄	C₂⁴
prime radical	C₂²×#41	C₂³
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

874 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:

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

Polycyclic presentation:

2^9|a^2=ef,b^2=f,c^2=i,d^2=ef,e^2=i,h^2=i,b^a=be,c^a=cf,d^a=dh,e^a=ei,f^a=fi,g^a=gi,h^a=hi,c^b=cg,d^c=de,e^c=ei,h^c=hi,h^d=hi

Conjugacy classes & character table

56 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 366908);