#146445

order	512
nilpotency class	4
isoclinism	isoclinic class of #13074
automorphism	?
homology	Abelianization (H₁) = C₂×C₄² · Schur multiplier (H₂) = C₂⁵×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×#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) = 128

A minimal faithful permutation representation — 3 generators on 36 points:

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

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=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	#72240706, #72240717, #72240739, …
maximal normal subgroup of	#72240706, #72240717, #72240739, …
maximal quotient of	#72240713, #72265362, #72307630

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 53641);