#233845

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

1176 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:

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

Polycyclic presentation:

2^9|a^2=hi,c^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=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, 141041);