#147363

order	512
nilpotency class	3
isoclinism	isoclinic class of #1913
automorphism	?
homology	Abelianization (H₁) = C₂×C₄×C₈ · Schur multiplier (H₂) = C₂⁴×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₂×C₄	#386
commutator	C₂³	C₂×C₄×C₈
Frattini, cosocle	C₂³×C₈	C₂³
socle	C₂²	#1869
prime radical	C₂⁵	C₂×C₈
LCS	G ▷ C₂³ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₄ ◁ C₂³×C₈ ◁ G
LECS	G ▷ C₂³×C₈ ▷ C₄ ▷ C₂ ▷ C₁

742 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

116 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72224874, #72224931, #72224933, …
maximal normal subgroup of	#72224874, #72224931, #72224933, …
maximal quotient of	#72224867, #72224871

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 54559);