#146449

order	512
nilpotency class	4
isoclinism	isoclinic class of #13077
automorphism	?
homology	Abelianization (H₁) = C₂×C₄² · Schur multiplier (H₂) = C₂⁶

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×#41	D₄
LCS	G ▷ C₂×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂³×C₄ ◁ C₂²×#131 ◁ G
LECS	G ▷ C₂³×C₈ ▷ C₄ ▷ C₂ ▷ C₁

892 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)(7,17)(8,14)(9,23)(10,27)(11,16)(12,18)(13,25)(15,24)(19,31)(20,33)(21,29)(22,28)(26,32)(30,36)(34,35);(5,7,9,20,10,21,8,22,11,19,25,36,24,35,26,18)(6,12,14,29,15,30,23,17,16,28,32,34,27,33,13,31);(5,6)(7,12)(8,23)(9,14)(10,15)(11,16)(13,26)(17,22)(18,31)(19,28)(20,29)(21,30)(24,27)(25,32)(33,35)(34,36)

Polycyclic presentation:

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

Conjugacy classes & character table

92 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72240739, #72240800, #72240815, …
maximal normal subgroup of	#72240739, #72240800, #72240815, …
maximal quotient of	#72240745, #72308291

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 53645);