#235156

order	512
nilpotency class	3
isoclinism	isoclinic class of #235139
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	C₂×#121	C₂³
LCS	G ▷ C₂³×C₄ ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂² ▷ C₁

1241 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)(3,11)(4,7)(5,9)(6,10)(8,13)(12,16)(14,15)(17,18)(19,29)(20,34)(21,27)(22,26)(23,28)(24,31)(25,36)(30,41)(32,40)(33,39)(35,43)(37,42)(38,45)(44,46)(47,48);(2,7)(4,5)(8,15)(9,10)(11,16)(12,13)(17,19,23,33)(18,24,28,40)(20,30,35,45)(21,32,22,31)(25,41,42,38)(26,39,27,29)(34,44,43,48)(36,47,37,46);(1,3,6,14)(2,8,10,16)(4,13,5,12)(7,15,9,11)(17,20,23,35)(18,25,28,42)(19,30,33,45)(21,36,22,37)(24,38,40,41)(26,34,27,43)(29,44,39,48)(31,46,32,47);(1,4,6,5)(2,7,10,9)(3,12,14,13)(8,11,16,15)(20,35)(25,42)(30,45)(34,43)(36,37)(38,41)(44,48)(46,47)

Polycyclic presentation:

2^9|b^2=hi,c^2=h,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=di,e^b=eh,f^b=fh,d^c=dh,e^c=ei,f^c=fh

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, 142352);