#250756

order	512
nilpotency class	3
isoclinism	isoclinic class of #250647
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	#61414	C₂
LCS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

1040 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)(3,8,5,7)(4,12,13,15)(9,14,16,11)(17,18)(19,29)(20,34)(21,27)(22,25)(23,28)(24,30)(26,36)(31,39)(32,41)(33,40)(35,43)(37,44)(38,42)(45,47)(46,48);(2,7)(3,5)(4,13)(8,10)(9,15)(12,16)(17,19)(18,24)(20,35)(21,32)(22,30)(23,33)(25,40)(26,43)(27,29)(28,41)(31,38)(34,48)(36,47)(37,46)(39,44)(42,45);(1,3,6,5)(2,8,10,7)(4,11,13,14)(9,15,16,12)(18,25)(19,30)(21,22)(26,42)(27,28)(29,40)(31,46)(32,33)(34,44)(35,47)(36,37)(45,48);(1,4,6,13)(2,9,10,16)(3,11,5,14)(7,12,8,15)(17,20,23,38)(18,26,28,44)(19,31,33,35)(21,37,22,36)(24,39,41,43)(25,42,27,34)(29,45,40,48)(30,46,32,47)

Polycyclic presentation:

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

Conjugacy classes & character table

50 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 157952);