#177758

order	512
nilpotency class	3
isoclinism	isoclinic class of #177733
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	C₂×#127	D₄
LCS	G ▷ C₂×C₄² ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₄³ ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂³ ▷ C₁

886 conjugacy classes of subgroups; only the 140 normal subgroups are shown.

normal subgroups only

This group has 886 conjugacy classes of subgroups — too many to draw. Only its 140 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:

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

Polycyclic presentation:

2^9|a^2=h,b^2=g,d^2=gh,e^2=g,f^2=i,b^a=be,c^a=cf,d^a=di,e^a=eg,f^a=fi,d^b=dgh,e^b=eg,f^b=fh,e^c=eh,f^c=fi

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, 84954);