#464716

order	512
nilpotency class	3
isoclinism	isoclinic class of #454045
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁴×C₄²

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	C₂×#120
commutator, Frattini, cosocle	C₂⁵	C₂⁴
prime radical	C₂×#534	C₂²
LCS	G ▷ C₂⁵ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂⁴×C₄ ◁ G
LECS	G ▷ C₂⁵ ▷ C₂ ▷ C₁

1850 conjugacy classes of subgroups; only the 210 normal subgroups are shown.

normal subgroups only

This group has 1850 conjugacy classes of subgroups — too many to draw. Only its 210 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:

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

Polycyclic presentation:

2^9|a^2=fi,b^2=eg,d^2=e,b^a=be,c^a=cf,d^a=dh,h^a=hi,c^b=cg,d^c=de,h^c=hi,f^d=fi

Conjugacy classes & character table

62 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 371912);