#233915

order	512
nilpotency class	3
isoclinism	isoclinic class of #233907
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	C₂×#532	C₂²
LCS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

1216 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:

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

Polycyclic presentation:

2^9|a^2=i,b^2=h,d^2=h,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=di,e^b=eh,f^b=fh,g^b=ghi,d^c=dh,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, 141111);