#235302

order	512
nilpotency class	3
isoclinism	isoclinic class of #235291
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

1240 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)(7,11)(12,16)(13,15)(17,18)(19,29)(20,33)(21,27)(22,26)(23,28)(24,31)(25,35)(30,38)(32,43)(34,42)(36,41)(37,39)(40,46)(44,45)(47,48);(1,3,6,13)(2,7,9,16)(4,11,5,12)(8,15,14,10)(17,19,23,32)(18,24,28,39)(20,30,36,46)(21,37,22,31)(25,40,42,38)(26,43,27,29)(33,44,41,48)(34,47,35,45);(1,4,6,5)(2,8,9,14)(3,11,13,12)(7,15,16,10)(17,20,23,36)(18,25,28,42)(19,30,32,46)(21,35,22,34)(24,38,39,40)(26,33,27,41)(29,44,43,48)(31,47,37,45);(1,4,6,5)(2,8,9,14)(3,12,13,11)(7,10,16,15)

Polycyclic presentation:

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

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, 142498);