#250481

order	512
nilpotency class	3
isoclinism	isoclinic class of #250455
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
LCS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

1046 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)(4,7)(5,9)(6,11)(8,14)(12,16)(13,15)(21,33)(25,34)(27,40)(28,42)(30,35)(32,44)(36,47)(39,45);(1,3)(2,6)(4,13)(5,12)(7,10)(8,16)(9,15)(11,14)(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);(6,15)(10,16)(17,19,22,29)(18,23,26,37)(20,28,31,42)(21,35,33,30)(24,36,38,47)(25,40,34,27)(32,46,44,43)(39,48,45,41);(2,7)(4,14)(6,10)(8,9)(11,13)(15,16)(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|b^2=i,c^2=i,d^2=i,e^2=h,g^2=i,b^a=be,c^a=cf,d^a=de,e^a=eh,g^a=ghi,c^b=cg,e^b=eh,f^b=fh,g^b=gi,e^c=ei,g^c=gi,e^d=eh,f^d=fi

Conjugacy classes & character table

44 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 157677);