#501249

order	512
nilpotency class	4
isoclinism	isoclinic class of #501248
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
LCS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂³×C₄ ◁ #2929 ◁ G
LECS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁

1057 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

56 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72323008
maximal normal subgroup of	#72323008

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 408445);