#146529

order	512
nilpotency class	4
isoclinism	isoclinic class of #2238
automorphism	?
homology	Abelianization (H₁) = C₂×C₄² · Schur multiplier (H₂) = C₂⁶

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₂×C₄	#481
commutator	C₂×C₈	C₂×C₄²
Frattini, cosocle	C₂³×C₈	C₂³
socle	C₂²	#1853
prime radical	C₂⁵	SD₁₆
LCS	G ▷ C₂×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₄ ◁ C₂³×C₄ ◁ C₂³×M₁₆ ◁ G
LECS	G ▷ C₂³×C₈ ▷ C₄ ▷ C₂ ▷ C₁

1018 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)(5,6,9,14,11,15,10,26)(7,16,19,29,18,33,20,36)(8,21,24,30,23,32,25,13)(12,22,28,17,27,35,31,34);(5,7)(6,12,15,27)(8,22)(9,20)(10,19)(11,18)(13,29,30,36)(14,31,26,28)(16,32,33,21)(17,25)(23,35)(24,34);(8,23)(13,30)(16,33)(17,34)(21,32)(22,35)(24,25)(29,36)

Polycyclic presentation:

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

Conjugacy classes & character table

80 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72215369, #72215373, #72216636, …
maximal normal subgroup of	#72215369, #72215373, #72216636, …
maximal quotient of	#72215302

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 53725);