#201039

order	512
nilpotency class	3
isoclinism	isoclinic class of #200914
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂⁵×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	C₂×#386
commutator	C₄²	C₂³×C₄
Frattini, cosocle	C₂×C₄²	C₂⁴
prime radical	C₂×#3153	C₂
LCS	G ▷ C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

1452 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 40
transitive degree	μ_t(G) = 64

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

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

Polycyclic presentation:

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

Conjugacy classes & character table

68 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72286262, #72286270, #72286278, …
maximal normal subgroup of	#72286262, #72286270, #72286278, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 108235);