#147132

order	512
nilpotency class	3
isoclinism	isoclinic class of #2031
automorphism	?
homology	Abelianization (H₁) = C₂×C₄² · Schur multiplier (H₂) = C₂⁴

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₂×C₈	#124
commutator	C₂²×C₄	C₂×C₄²
Frattini, cosocle	C₂³×C₈	C₂³
socle	C₂²	#1985
prime radical	C₂⁴	C₂×M₁₆
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₈ ◁ C₂³×C₈ ◁ G
LECS	G ▷ C₂³×C₈ ▷ C₄ ▷ C₂ ▷ C₁

544 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)(7,11,16,24)(8,19)(9,14)(10,12)(13,21)(15,18,26,23)(17,32,31,25)(20,29)(22,27)(28,35)(30,36,33,34);(5,7,8,17,9,18,20,34,10,16,21,31,22,23,35,36)(6,11,13,25,14,26,28,33,12,24,19,32,27,15,29,30);(2,4)(5,6)(7,15,16,26)(8,19)(9,14)(10,12)(11,23,24,18)(13,21)(17,33,31,30)(20,29)(22,27)(25,34,32,36)(28,35)

Polycyclic presentation:

2^9|a^2=f,b^2=g,c^2=f,g^2=h,h^2=i,b^a=bd,c^a=ce,d^a=di,g^a=gi,c^b=ch,d^b=di,e^b=ei,f^b=fi,d^c=di,g^c=gi

Conjugacy classes & character table

104 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72224874, #72226807, #72226810, …
maximal normal subgroup of	#72224874, #72226807, #72226810, …
maximal quotient of	#72224870

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 54328);