#249237

order	512
nilpotency class	3
isoclinism	isoclinic class of #249199
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁶

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	C₂×#120
commutator, Frattini, cosocle	C₂³×C₄	C₂⁴
prime radical	C₂⁴	C₂²×D₄
LCS	G ▷ C₂³×C₄ ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂² ▷ C₁

1122 conjugacy classes of subgroups; only the 166 normal subgroups are shown.

normal subgroups only

This group has 1122 conjugacy classes of subgroups — too many to draw. Only its 166 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) = 52
transitive degree	μ_t(G) = 128

A minimal faithful permutation representation — 4 generators on 52 points:

(7,14)(8,17)(9,12)(11,19)(15,20)(16,18)(21,22,26,30)(23,27,33,40)(24,28,35,42)(25,36,37,29)(31,43,45,38)(32,46,47,39)(34,41,48,49)(44,51,52,50);(1,2)(3,4)(5,6,10,13)(7,11,16,20)(8,9,17,12)(14,15,18,19)(21,23)(22,27)(24,31)(25,32)(26,33)(28,38)(29,39)(30,40)(34,44)(35,45)(36,46)(37,47)(41,50)(42,43)(48,52)(49,51);(2,4)(5,7,10,16)(6,11,13,20)(8,15,17,19)(9,18,12,14)(21,24)(22,28)(23,31)(25,34)(26,35)(27,38)(29,41)(30,42)(32,44)(33,45)(36,49)(37,48)(39,50)(40,43)(46,51)(47,52);(5,8,10,17)(6,12,13,9)(7,15,16,19)(11,14,20,18)(21,25,26,37)(22,29,30,36)(23,32,33,47)(24,34,35,48)(27,39,40,46)(28,41,42,49)(31,44,45,52)(38,50,43,51)

Polycyclic presentation:

2^9|a^2=hi,b^2=i,c^2=i,d^2=h,f^2=i,b^a=be,c^a=cf,d^a=deh,f^a=fi,c^b=cg,d^b=di,f^b=fh,e^c=eh,f^c=fi,f^d=fh

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, 156433);