#410815

order	512
nilpotency class	3
isoclinism	isoclinic class of #393877
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	C₂²×#121	C₂²
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂⁴×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

1544 conjugacy classes of subgroups; only the 242 normal subgroups are shown.

normal subgroups only

This group has 1544 conjugacy classes of subgroups — too many to draw. Only its 242 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) = 32
transitive degree	μ_t(G) = 128

A minimal faithful permutation representation — 4 generators on 32 points:

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

Polycyclic presentation:

2^9|b^2=gh,c^2=g,d^2=i,e^2=i,b^a=be,c^a=cf,d^a=dh,e^a=ei,c^b=cg,e^b=ei,f^b=fi,e^c=ei

Conjugacy classes & character table

74 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 318011);