#409965

order	512
nilpotency class	3
isoclinism	isoclinic class of #393929
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, prime radical, cosocle	C₂⁵	C₂⁴
LCS	G ▷ C₂⁵ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂⁴×C₄ ◁ G
LECS	G ▷ C₂⁵ ▷ C₂ ▷ C₁

1826 conjugacy classes of subgroups; only the 258 normal subgroups are shown.

normal subgroups only

This group has 1826 conjugacy classes of subgroups — too many to draw. Only its 258 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:

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

Polycyclic presentation:

2^9|a^2=i,b^2=h,c^2=g,d^2=i,b^a=be,c^a=cf,d^a=dh,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, 317161);