#147310

order	512
nilpotency class	4
isoclinism	isoclinic class of #2230
automorphism	?
homology	Abelianization (H₁) = C₂×C₄² · Schur multiplier (H₂) = C₂³×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₂×C₄	#476
commutator	C₂×C₈	C₂×C₄²
Frattini, cosocle	C₂³×C₈	C₂³
socle	C₂²	#1920
prime radical	C₂⁴	#137
LCS	G ▷ C₂×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₄ ◁ C₂³×C₄ ◁ C₂²×C₄×C₈ ◁ G
LECS	G ▷ C₂³×C₈ ▷ C₄ ▷ C₂ ▷ C₁

530 conjugacy classes of subgroups.

Full subgroup data →

Presentation

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

A minimal faithful permutation representation — 3 generators on 36 points:

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

Polycyclic presentation:

2^9|a^2=f,b^2=g,c^2=fh,d^2=h,e^2=hi,g^2=i,h^2=i,b^a=bd,c^a=ce,d^a=dh,e^a=ehi,h^a=hi,d^b=dhi,e^b=ehi,f^b=fi,h^b=hi,d^c=dhi,e^c=ehi,f^c=fi,h^c=hi

Conjugacy classes & character table

92 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72314209, #72314250, #72314268
maximal normal subgroup of	#72314209, #72314250, #72314268

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 54506);