#233851

order	512
nilpotency class	3
isoclinism	isoclinic class of #233836
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	#3444	C₂²
LCS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

1135 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 48
transitive degree	μ_t(G) = 64

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

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

Polycyclic presentation:

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

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, 141047);