#147662

order	512
nilpotency class	4
isoclinism	isoclinic class of #13310
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄ · Schur multiplier (H₂) = C₂⁶

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#1999
commutator	C₂²×C₈	C₂²×C₄
Frattini, cosocle	C₂³×C₈	C₂³
prime radical	C₂×#3058	C₂
LCS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂³×C₄ ◁ C₂²×#131 ◁ G
LECS	G ▷ C₂³×C₈ ▷ C₄ ▷ C₂ ▷ C₁

1104 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)(5,6,9,14,10,15,26,16)(7,17,19,31,21,32,20,12)(8,22,24,13,25,11,23,28)(18,33,34,27,36,30,35,29);(6,11)(8,23)(9,26)(12,29)(13,14)(15,22)(16,28)(17,30)(18,34)(19,20)(24,25)(27,31)(32,33)(35,36);(2,4)(5,7)(6,12)(8,18)(9,20)(10,21)(11,27)(13,30)(14,32)(15,31)(16,17)(19,26)(22,29)(23,34)(24,35)(25,36)(28,33)

Polycyclic presentation:

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

Conjugacy classes & character table

68 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72231521, #72231532, #72231546, …
maximal normal subgroup of	#72231521, #72231532, #72231546, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 54858);