#205488

order	512
nilpotency class	3
isoclinism	isoclinic class of #19162
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂⁵×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₂×C₄	C₂×#121
commutator	C₄²	C₂³×C₄
Frattini, cosocle	C₂×C₄²	C₂⁴
socle	C₂²	C₂×#390
prime radical	#60639	C₂
LCS	G ▷ C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂×C₄ ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

1321 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 40
transitive degree	μ_t(G) = 128

A minimal faithful permutation representation — 4 generators on 40 points:

(1,2,3,6,4,7,8,5)(9,10,14,18,15,19,13,29)(11,20,24,31,22,34,23,16)(12,25,28,32,26,39,27,17)(21,35,37,40,38,30,36,33);(2,5)(3,8)(6,7)(9,11)(10,16,19,31)(12,21)(13,23)(14,24)(15,22)(17,30,32,35)(18,20,29,34)(25,33,39,40)(26,38)(27,36)(28,37);(9,12,13,27,15,26,14,28)(10,17,18,25,19,32,29,39)(11,21,23,36,22,38,24,37)(16,30,20,33,31,35,34,40);(11,22)(12,26)(16,31)(17,32)(20,34)(23,24)(25,39)(27,28)

Polycyclic presentation:

2^9|a^2=g,b^2=egi,c^2=f,e^2=hi,f^2=h,g^2=hi,b^a=be,c^a=cf,e^a=eh,f^a=fh,d^b=dh,e^b=ei,g^b=gi,d^c=dh

Conjugacy classes & character table

86 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72285399, #72286268, #72286509
maximal normal subgroup of	#72285399, #72286268, #72286509
maximal quotient of	#72285524, #72319589

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 112684);