#501258

order	512
nilpotency class	4
isoclinism	isoclinic class of #501256
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁶

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	C₂×#447
commutator, Frattini, cosocle	C₂²×C₈	C₂⁴
prime radical	#61030	C₂
LCS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂³×C₄ ◁ C₂×#435 ◁ G
LECS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁

1095 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:

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

Polycyclic presentation:

2^9|a^2=f,c^2=i,e^2=h,h^2=i,b^a=be,c^a=cf,d^a=de,e^a=eh,h^a=hi,c^b=cg,e^b=ehi,f^b=fi,h^b=hi,e^c=ei,e^d=ehi,f^d=fi,h^d=hi

Conjugacy classes & character table

56 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72251066, #72252778, #72252788
maximal normal subgroup of	#72251066, #72252778, #72252788

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 408454);