#250769

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	#3497	C₂²
LCS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

1039 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:

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

Polycyclic presentation:

2^9|a^2=hi,b^2=h,c^2=i,d^2=i,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, 157965);