#250605

order	512
nilpotency class	3
isoclinism	isoclinic class of #250519
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	#532	C₂³
LCS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

978 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)(3,12)(4,8)(5,9)(6,11)(7,13)(10,15)(14,16)(17,18,24,29)(19,25,32,21)(20,33,36,45)(22,28,23,27)(26,38,44,37)(30,43,46,48)(31,39,41,40)(34,42,47,35);(1,3,6,4)(2,7,11,16)(5,14,15,13)(8,9,12,10)(17,19)(18,25)(20,34)(21,29)(22,31)(23,41)(24,32)(26,43)(27,40)(28,39)(30,38)(33,35)(36,47)(37,46)(42,45)(44,48);(1,4,6,3)(2,8,11,12)(5,14,15,13)(7,9,16,10)(17,20)(18,26)(19,30)(21,35)(22,38)(23,37)(24,36)(25,42)(27,45)(28,33)(29,44)(31,47)(32,46)(34,41)(39,43)(40,48);(2,9)(5,15)(7,12)(8,16)(10,11)(13,14)(17,21,24,25)(18,19,29,32)(20,35,36,42)(22,40,23,39)(26,30,44,46)(27,41,28,31)(33,47,45,34)(37,43,38,48)

Polycyclic presentation:

2^9|a^2=i,b^2=h,c^2=h,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,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

56 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 157801);