#244654

order	512
nilpotency class	3
isoclinism	isoclinic class of #244583
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

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

normal subgroups only

This group has 931 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)(3,10)(4,11)(5,7)(6,9)(8,15)(12,13)(14,16)(17,18,22,26)(19,27,30,41)(20,31,33,42)(21,34,37,25)(23,35,38,29)(24,36,40,32)(28,39,45,47)(43,46,48,44);(2,7)(3,11)(5,15)(8,9)(10,13)(12,16)(17,19,22,30)(18,23,26,38)(20,28,33,45)(21,35,37,29)(24,39,40,47)(25,41,34,27)(31,43,42,48)(32,46,36,44);(1,3,6,12)(2,4,9,14)(5,16,15,11)(7,10,8,13)(23,38)(27,41)(39,47)(43,48);(1,4,6,14)(2,3,9,12)(5,13,15,10)(7,11,8,16)(17,20,22,33)(18,24,26,40)(19,28,30,45)(21,36,37,32)(23,39,38,47)(25,42,34,31)(27,43,41,48)(29,46,35,44)

Polycyclic presentation:

2^9|a^2=hi,b^2=hi,c^2=i,d^2=h,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=ghi,c^b=cg,e^b=eh,f^b=fh,g^b=gi,e^c=ei,f^c=fi,g^c=gi,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, 151850);