#147898

order	512
nilpotency class	4
isoclinism	isoclinic class of #13417
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄ · Schur multiplier (H₂) = C₂⁶

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#2041
commutator	C₂²×C₈	C₂²×C₄
Frattini, cosocle	C₂³×C₈	C₂³
prime radical	C₂×#3155	C₂
LCS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂³×C₄ ◁ C₂×#405 ◁ G
LECS	G ▷ C₂³×C₈ ▷ C₄ ▷ C₂ ▷ C₁

1198 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)(5,6,9,17,11,16,10,28)(7,18,21,35,22,33,29,34)(8,23,26,15,27,13,25,14)(12,24,31,20,30,36,32,19);(2,4)(5,7)(6,12)(8,24)(9,21)(10,29)(11,22)(13,33)(14,34)(15,35)(16,30)(17,31)(18,23)(19,25)(20,26)(27,36)(28,32);(6,13)(8,25)(9,10)(12,18)(14,28)(15,17)(16,23)(19,24)(20,36)(21,29)(26,27)(30,33)(31,34)(32,35)

Polycyclic presentation:

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

Conjugacy classes & character table

68 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72224865, #72226810, #72226816, …
maximal normal subgroup of	#72224865, #72226810, #72226816, …
maximal quotient of	#72224862, #72238787, #72280166

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 55094);