#146446

order	512
nilpotency class	4
isoclinism	isoclinic class of #13074
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

884 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)(7,16)(8,20)(9,24)(10,15)(11,17)(12,22)(13,28)(14,21)(18,26)(19,35)(23,29)(25,30)(27,36)(31,34)(32,33);(5,7,8,17,21,33,28,36,10,18,23,30,9,19,22,34)(6,11,13,26,24,31,20,32,15,25,12,16,14,27,29,35);(5,6)(7,11)(8,13)(9,14)(10,15)(12,23)(16,30)(17,26)(18,25)(19,27)(20,28)(21,24)(22,29)(31,33)(32,36)(34,35)

Polycyclic presentation:

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

Conjugacy classes & character table

92 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72240708, #72240716, #72240739, …
maximal normal subgroup of	#72240708, #72240716, #72240739, …
maximal quotient of	#72240713, #72280403, #72308322

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 53642);