#148433

order	512
nilpotency class	4
isoclinism	isoclinic class of #13340
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄ · Schur multiplier (H₂) = C₂⁵

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₂×C₄	#493
commutator	C₂²×C₈	C₂²×C₄
Frattini, cosocle	C₂³×C₈	C₂³
socle	C₂²	#2093
prime radical	C₂²×#135	C₂²
LCS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₄ ◁ C₂³×C₄ ◁ C₂²×#97 ◁ G
LECS	G ▷ C₂³×C₈ ▷ C₄ ▷ C₂ ▷ C₁

674 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:

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

Polycyclic presentation:

2^9|a^2=g,b^2=dh,d^2=hi,f^2=hi,g^2=i,h^2=i,b^a=bd,c^a=ce,d^a=dh,f^a=fhi,h^a=hi,c^b=cf,e^b=ei,d^c=dhi,f^c=fh,h^c=hi

Conjugacy classes & character table

68 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72250044, #72250050, #72250053, …
maximal normal subgroup of	#72250044, #72250050, #72250053, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 55629);