#418853

order	512
nilpotency class	3
isoclinism	isoclinic class of #418851
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁷

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₄	#2679
commutator, Frattini, cosocle	C₂³×C₄	C₂⁴
socle	C₂	#15266
prime radical	#60387	C₂
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₄ ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

1900 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 32
transitive degree	μ_t(G) = 32

A minimal faithful permutation representation — 4 generators on 32 points:

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

Polycyclic presentation:

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

Conjugacy classes & character table

62 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72263012, #72267112, #72318925, …
maximal normal subgroup of	#72263012, #72267112, #72318925, …
maximal quotient of	#72262937

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 326049);