#493553

order	512
nilpotency class	4
isoclinism	isoclinic class of #33539
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂⁸

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	C₂×#447
commutator	C₂×C₈	C₂³×C₄
Frattini, cosocle	C₂²×C₈	C₂⁴
prime radical	C₂×#3584	C₂
LCS	G ▷ C₂×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂³×C₄ ◁ C₂²×#131 ◁ G
LECS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁

3140 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 36
transitive degree	μ_t(G) = 128

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

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

Polycyclic presentation:

2^9|a^2=g,d^2=i,e^2=hi,f^2=i,g^2=i,h^2=i,b^a=be,c^a=cf,d^a=di,e^a=eh,h^a=hi,e^b=eh,h^b=hi,f^c=fi,g^c=gi

Conjugacy classes & character table

80 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72321321, #72321366, #72321415, …
maximal normal subgroup of	#72321321, #72321366, #72321415, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 400749);