#146760

order	512
nilpotency class	4
isoclinism	isoclinic class of #12948
automorphism	?
homology	Abelianization (H₁) = C₂×C₄² · Schur multiplier (H₂) = C₂⁶

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#1864
commutator	C₂×C₈	C₂×C₄²
Frattini, cosocle	C₂³×C₈	C₂³
prime radical	C₂²×#135	C₂²
LCS	G ▷ C₂×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂³×C₄ ◁ C₂×#405 ◁ G
LECS	G ▷ C₂³×C₈ ▷ C₄ ▷ C₂ ▷ C₁

950 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)(6,13,16,15)(7,18,19,35)(8,21)(9,24)(10,11)(14,30)(17,25,27,22)(20,26,28,23)(29,36)(31,34);(5,6,10,17,12,16,11,27)(7,14,20,32,19,30,28,33)(8,22,24,15,9,25,21,13)(18,29,23,31,35,34,26,36);(5,7)(6,14)(8,23)(9,26)(10,20)(11,28)(12,19)(13,29)(15,34)(16,30)(17,32)(18,21)(22,31)(24,35)(25,36)(27,33)

Polycyclic presentation:

2^9|a^2=f,b^2=g,d^2=h,e^2=h,g^2=i,h^2=i,b^a=bd,c^a=ce,d^a=dh,e^a=eh,g^a=gi,h^a=hi,d^b=dh,e^b=eh,f^b=fi,h^b=hi,d^c=dh,e^c=ehi,f^c=fi,h^c=hi

Conjugacy classes & character table

80 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72250498, #72250502, #72250510, …
maximal normal subgroup of	#72250498, #72250502, #72250510, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 53956);