#249830

order	512
nilpotency class	3
isoclinism	isoclinic class of #249799
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	#60725	C₂
LCS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

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

normal subgroups only

This group has 994 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) = 128

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

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

Polycyclic presentation:

2^9|a^2=i,b^2=i,c^2=h,e^2=h,f^2=hi,g^2=h,b^a=be,c^a=cf,d^a=de,e^a=eh,f^a=fhi,c^b=cg,e^b=eh,f^b=fi,g^b=gh,e^c=ei,f^c=fhi,g^c=gh,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, 157026);