#413855

order	512
nilpotency class	3
isoclinism	isoclinic class of #393929
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁵

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	C₂×#120
commutator, Frattini, prime radical, cosocle	C₂⁵	C₂⁴
LCS	G ▷ C₂⁵ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂⁴×C₄ ◁ G
LECS	G ▷ C₂⁵ ▷ C₂ ▷ C₁

1480 conjugacy classes of subgroups; only the 270 normal subgroups are shown.

normal subgroups only

This group has 1480 conjugacy classes of subgroups — too many to draw. Only its 270 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) = 28
transitive degree	μ_t(G) = 128

A minimal faithful permutation representation — 4 generators on 28 points:

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

Polycyclic presentation:

2^9|a^2=e,b^2=g,c^2=g,d^2=hi,b^a=be,c^a=cf,d^a=dh,f^a=fi,c^b=cg,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, 321051);