#235178

order	512
nilpotency class	3
isoclinism	isoclinic class of #235159
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	C₂³×C₄	C₂⁴
prime radical	#3186	C₂²
LCS	G ▷ C₂³×C₄ ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂² ▷ C₁

1233 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)(3,12)(4,9)(5,10)(6,11)(7,14)(8,13)(15,16)(17,18)(19,29)(20,34)(21,27)(22,26)(23,28)(24,32)(25,35)(30,41)(31,40)(33,39)(36,43)(37,42)(38,45)(44,46)(47,48);(1,3,6,15)(2,7,11,8)(4,13,5,14)(9,12,10,16)(17,19)(18,24)(20,30)(21,33)(22,32)(23,31)(25,41)(26,40)(27,29)(28,39)(34,44)(35,47)(36,45)(37,46)(38,42)(43,48);(1,3)(2,8)(4,13)(5,14)(6,15)(7,11)(9,16)(10,12)(17,20)(18,25)(19,30)(21,35)(22,37)(23,36)(24,38)(26,34)(27,43)(28,42)(29,44)(31,45)(32,46)(33,47)(39,41)(40,48);(1,4,6,5)(2,9,11,10)(3,13,15,14)(7,12,8,16)(19,31)(24,39)(29,40)(30,45)(32,33)(38,41)(44,48)(46,47)

Polycyclic presentation:

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

Conjugacy classes & character table

68 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 142374);