#450385

order	512
nilpotency class	3
isoclinism	isoclinic class of #418851
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁵

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₄	#2679
commutator, Frattini, cosocle	C₂³×C₄	C₂⁴
socle	C₂	#15966
prime radical	#3449	C₂²
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₄ ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

1050 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

62 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72319908, #72319954, #72320075, …
maximal normal subgroup of	#72319908, #72319954, #72320075, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 357581);