#353434

order	512
nilpotency class	3
isoclinism	isoclinic class of #3026
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄² · Schur multiplier (H₂) = C₂⁹

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₂×C₄	C₂×#120
commutator	C₂³	C₂²×C₄²
Frattini, cosocle	C₂³×C₄	C₂⁴
socle	C₂²	C₂×#385
prime radical	C₂³×#41	C₂²
LCS	G ▷ C₂³ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₄ ◁ C₂⁴×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

4038 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)(5,6,9,15,11,14,10,16)(7,17,20,30,19,12,24,26)(8,13,22,28,21,27,23,29)(18,25,33,35,32,31,34,36);(5,7,10,20,11,19,9,24)(6,12,16,26,14,17,15,30)(8,18,23,33,21,32,22,34)(13,25,29,35,27,31,28,36);(2,4)(5,8)(6,13)(7,18)(9,22)(10,23)(11,21)(12,25)(14,27)(15,28)(16,29)(17,31)(19,32)(20,33)(24,34)(26,35)(30,36);(6,14)(12,17)(13,27)(15,16)(25,31)(26,30)(28,29)(35,36)

Polycyclic presentation:

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

Conjugacy classes & character table

116 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72251307, #72251319, #72251323, …
maximal normal subgroup of	#72251307, #72251319, #72251323, …
maximal quotient of	#72279363, #72279851, #72280607, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 260630);