#147383

order	512
nilpotency class	3
isoclinism	isoclinic class of #11000
automorphism	?
homology	Abelianization (H₁) = C₂×C₄² · Schur multiplier (H₂) = C₂⁵

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₂×C₄	#390
commutator	C₂²×C₄	C₂×C₄²
Frattini, cosocle	C₂³×C₈	C₂³
socle	C₂²	#1955
prime radical	C₂⁵	M₁₆
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₄ ◁ C₂³×C₈ ◁ G
LECS	G ▷ C₂³×C₈ ▷ C₄ ▷ C₂ ▷ C₁

716 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)(5,6,10,16,11,15,26,31,12,8,14,24,9,23,25,36)(7,17,20,32,21,22,28,35,19,33,30,34,18,13,27,29);(2,4)(5,7)(6,13,8,22)(9,18)(10,20)(11,21)(12,19)(14,30)(15,17,23,33)(16,29,24,35)(25,27)(26,28)(31,32,36,34);(5,8,10,16,11,23,26,31,12,6,14,24,9,15,25,36)(7,17,30,32,21,22,27,35,19,33,20,34,18,13,28,29)

Polycyclic presentation:

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

Conjugacy classes & character table

92 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72224874, #72224936, #72224939
maximal normal subgroup of	#72224874, #72224936, #72224939
maximal quotient of	#72224868

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 54579);