#147410

order	512
nilpotency class	4
isoclinism	isoclinic class of #13077
automorphism	?
homology	Abelianization (H₁) = C₂×C₄² · Schur multiplier (H₂) = C₂⁴×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#1893
commutator	C₂×C₈	C₂×C₄²
Frattini, cosocle	C₂³×C₈	C₂³
prime radical	C₂²×#135	C₂²
LCS	G ▷ C₂×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂³×C₄ ◁ C₂×#405 ◁ G
LECS	G ▷ C₂³×C₈ ▷ C₄ ▷ C₂ ▷ C₁

734 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:

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

Polycyclic presentation:

2^9|a^2=f,b^2=g,c^2=ef,d^2=h,g^2=hi,h^2=i,b^a=bd,c^a=ce,d^a=dh,g^a=ghi,h^a=hi,d^b=di,e^b=ei,f^b=fi,d^c=di

Conjugacy classes & character table

92 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72224893, #72224933, #72224942, …
maximal normal subgroup of	#72224893, #72224933, #72224942, …
maximal quotient of	#72224888

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 54606);