#235567

order	512
nilpotency class	3
isoclinism	isoclinic class of #235558
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁶

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	C₂×#392
commutator, Frattini, cosocle	C₂×C₄²	C₂⁴
prime radical	C₂²×#41	D₄
LCS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

1207 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)(3,10)(4,8)(5,14)(6,9)(7,13)(11,15)(12,16)(17,18,23,28)(19,29,32,40)(20,25,36,43)(21,27,22,26)(24,33,39,31)(30,45,46,41)(34,48,47,38)(35,44,37,42);(7,15)(10,16)(17,19)(18,24)(20,34)(21,33)(22,31)(23,32)(25,41)(26,40)(27,29)(28,39)(30,35)(36,47)(37,46)(38,42)(43,45)(44,48);(1,3)(2,7)(4,13)(5,11)(6,12)(8,16)(9,15)(10,14)(17,20,23,36)(18,25,28,43)(19,30,32,46)(21,35,22,37)(24,38,39,48)(26,42,27,44)(29,41,40,45)(31,34,33,47);(1,4,6,5)(2,8,9,14)(3,11,12,13)(7,10,15,16)

Polycyclic presentation:

2^9|a^2=h,c^2=h,d^2=i,e^2=h,f^2=i,g^2=h,b^a=be,c^a=cf,e^a=eh,f^a=fi,g^a=ghi,c^b=cg,e^b=eh,f^b=fh,g^b=gh,d^c=di,e^c=ei,f^c=fi,g^c=gh

Conjugacy classes & character table

56 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 142763);