#245335

order	512
nilpotency class	3
isoclinism	isoclinic class of #245303
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

968 conjugacy classes of subgroups; only the 146 normal subgroups are shown.

normal subgroups only

This group has 968 conjugacy classes of subgroups — too many to draw. Only its 146 normal subgroups are shown.

Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.

Full subgroup data →

Presentation

minimal degree	μ(G) = 48
transitive degree	μ_t(G) = 256

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

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

Polycyclic presentation:

2^9|a^2=hi,b^2=h,d^2=i,e^2=h,f^2=i,g^2=h,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=gh,e^c=ei,f^c=fi,g^c=gh,e^d=eh,f^d=fi

Conjugacy classes & character table

47 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 152531);