#250750

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	#3150	C₂²
LCS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

997 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)(18,24)(19,29)(20,34)(21,22)(27,28)(30,40)(31,45)(32,33)(36,37)(38,42)(39,44)(46,48);(2,7)(3,5)(4,13)(8,10)(9,15)(12,16)(17,18)(19,30)(20,35)(21,27)(22,24)(23,28)(25,29)(26,37)(31,46)(32,41)(33,40)(34,42)(36,38)(39,47)(43,48)(44,45);(1,3,6,5)(2,8,10,7)(4,11,13,14)(9,15,16,12)(17,19,23,33)(18,25,28,41)(20,31,37,45)(21,29,22,32)(24,30,27,40)(26,39,35,48)(34,47,36,43)(38,44,42,46);(1,4,6,13)(2,9,10,16)(3,11,5,14)(7,12,8,15)(17,20)(18,26)(19,31)(21,36)(22,34)(23,37)(24,38)(25,39)(27,42)(28,35)(29,43)(30,44)(32,47)(33,45)(40,46)(41,48)

Polycyclic presentation:

2^9|a^2=hi,c^2=h,d^2=hi,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, 157946);