#250381

order	512
nilpotency class	3
isoclinism	isoclinic class of #250327
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁶

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#2403
commutator, Frattini, cosocle	C₂×C₄²	C₂⁴
prime radical	#3515	C₂²
LCS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

939 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

47 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 157577);