#250454

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
LCS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

1013 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:

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

Polycyclic presentation:

2^9|a^2=hi,b^2=i,c^2=hi,d^2=hi,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, 157650);