#194337

order	512
nilpotency class	3
isoclinism	isoclinic class of #18866
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂⁶

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₂×C₄	C₂×#120
commutator	C₄²	C₂³×C₄
Frattini, cosocle	C₂×C₄²	C₂⁴
socle	C₂²	C₂×#386
prime radical	C₂⁴	#121
LCS	G ▷ C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂×C₄ ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

1142 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:

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

Polycyclic presentation:

2^9|a^2=g,b^2=eg,c^2=h,e^2=h,f^2=i,g^2=i,b^a=be,c^a=cf,d^a=di,d^b=dh,e^b=eh,f^b=fi,g^b=gh,e^c=ei,f^c=fi,g^c=gi

Conjugacy classes & character table

86 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 101533);