#196238

order	512
nilpotency class	3
isoclinism	isoclinic class of #1909
automorphism	?
homology	Abelianization (H₁) = C₂³×C₈ · Schur multiplier (H₂) = C₂⁷

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₂×C₄	#386
commutator	C₂×C₄	C₂³×C₈
Frattini, cosocle	C₂×C₄²	C₂⁴
socle	C₂²	C₂×#386
prime radical	C₂²×#41	C₂×C₄
LCS	G ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₄ ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

1369 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:

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

Polycyclic presentation:

2^9|a^2=g,b^2=eg,c^2=h,d^2=hi,e^2=h,g^2=i,b^a=be,c^a=cf,e^a=eh,f^a=fh,f^b=fh,e^c=eh,g^c=gh

Conjugacy classes & character table

116 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 103434);