#177729

order	512
nilpotency class	3
isoclinism	isoclinic class of #177385
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁴

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	C₂×#120
commutator, Frattini, cosocle	C₂×C₄²	C₂⁴
prime radical	C₂²×D₄	C₂×D₄
LCS	G ▷ C₂×C₄² ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₄³ ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂³ ▷ C₁

678 conjugacy classes of subgroups; only the 130 normal subgroups are shown.

normal subgroups only

This group has 678 conjugacy classes of subgroups — too many to draw. Only its 130 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) = 40
transitive degree	μ_t(G) = 256

A minimal faithful permutation representation — 4 generators on 40 points:

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

Polycyclic presentation:

2^9|a^2=hi,b^2=g,c^2=h,d^2=ghi,e^2=g,f^2=i,b^a=be,c^a=cf,d^a=dghi,e^a=eg,f^a=fi,e^b=eg,f^b=fh,d^c=dg,e^c=eh,f^c=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, 84925);