#193612

order	512
nilpotency class	3
isoclinism	isoclinic class of #23860
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂⁶

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₂²×C₄	C₂²×D₄
commutator	C₂²×C₄	C₂³×C₄
Frattini, cosocle	C₂×C₄²	C₂⁴
socle	C₂³	C₂×C₄×D₄
prime radical	C₂⁴	C₄×D₄
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂²×C₄ ◁ C₂×#431 ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

1098 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 52
transitive degree	μ_t(G) = 256

A minimal faithful permutation representation — 4 generators on 52 points:

(2,4)(5,6,8,11,9,12,15,18)(7,10,14,17,13,16,19,20)(21,22)(23,33,37,44)(24,38,41,46)(25,31)(26,32)(27,30)(28,35,43,36)(29,40,45,39)(34,48)(42,51)(47,49)(50,52);(1,2,3,4)(5,7,8,14,9,13,15,19)(6,10,11,17,12,16,18,20)(21,23,25,36,27,37,26,35)(22,28,31,44,30,43,32,33)(24,34,40,49,41,51,39,50)(29,42,38,52,45,48,46,47);(21,24,27,41)(22,29,30,45)(23,34,37,51)(25,40,26,39)(28,42,43,48)(31,38,32,46)(33,47,44,52)(35,50,36,49);(7,13)(10,16)(14,19)(17,20)(22,30)(28,43)(29,45)(31,32)(33,44)(38,46)(42,48)(47,52)

Polycyclic presentation:

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

Conjugacy classes & character table

92 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 100808);