#146450

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

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

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=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, #72240799, #72240816, …
maximal normal subgroup of	#72240739, #72240799, #72240816, …
maximal quotient of	#72240745, #72265374

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 53646);