#235392

order	512
nilpotency class	3
isoclinism	isoclinic class of #235381
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

1238 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)(3,8,13,16)(4,9,5,7)(11,14,12,15)(17,18,23,28)(19,29,32,41)(20,33,34,42)(21,27,22,26)(24,36,39,31)(25,37,40,35)(30,38,44,48)(43,45,47,46);(2,7)(4,5)(8,15)(9,10)(11,12)(14,16)(17,19)(18,24)(20,30)(21,36)(22,31)(23,32)(25,38)(26,41)(27,29)(28,39)(33,47)(34,44)(35,46)(37,45)(40,48)(42,43);(1,3,6,13)(2,8,10,16)(4,12,5,11)(7,14,9,15)(17,20,23,34)(18,25,28,40)(19,30,32,44)(21,35,22,37)(24,38,39,48)(26,33,27,42)(29,43,41,47)(31,45,36,46);(1,4,6,5)(2,9,10,7)(3,11,13,12)(8,14,16,15)(20,34)(25,40)(30,44)(33,42)(35,37)(38,48)(43,47)(45,46)

Polycyclic presentation:

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

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, 142588);