#245894

order	512
nilpotency class	3
isoclinism	isoclinic class of #245807
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

1132 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 48
transitive degree	μ_t(G) = 64

A minimal faithful permutation representation — 4 generators on 48 points:

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

Polycyclic presentation:

2^9|b^2=i,c^2=hi,d^2=i,e^2=h,f^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,e^c=ei,f^c=fi,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, 153090);