#403113

order	512
nilpotency class	3
isoclinism	isoclinic class of #394465
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁶

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₂×C₄	C₂×#124
commutator, Frattini, cosocle	C₂³×C₄	C₂⁴
socle	C₂²	#2398
prime radical	C₂×#575	C₂²
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₄ ◁ C₂³×C₈ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

1201 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 36
transitive degree	μ_t(G) = 64

A minimal faithful permutation representation — 4 generators on 36 points:

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

Polycyclic presentation:

2^9|a^2=fg,b^2=ef,d^2=ef,e^2=i,g^2=i,b^a=be,c^a=cf,d^a=dh,e^a=ei,f^a=fi,g^a=gi,c^b=cg,e^b=ei,f^b=fi,g^b=gi,g^c=gi

Conjugacy classes & character table

74 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72257146, #72257150, #72257156
maximal normal subgroup of	#72257146, #72257150, #72257156

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 310309);