#201804

order	512
nilpotency class	3
isoclinism	isoclinic class of #201717
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂⁶

Aliases

No aliases for this group.

Subgroups

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

1082 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 48
transitive degree	μ_t(G) = 128

A minimal faithful permutation representation — 4 generators on 48 points:

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

Polycyclic presentation:

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

Conjugacy classes & character table

62 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 109000);