#321412

order	512
nilpotency class	3
isoclinism	isoclinic class of #298661
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂³×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle, prime radical	C₂³	C₂×#127
commutator, Frattini, cosocle	C₂×C₄²	C₂⁴
LCS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₄³ ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

780 conjugacy classes of subgroups; only the 206 normal subgroups are shown.

normal subgroups only

This group has 780 conjugacy classes of subgroups — too many to draw. Only its 206 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) = 512

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

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

Polycyclic presentation:

2^9|a^2=fh,b^2=eg,c^2=i,d^2=e,f^2=h,g^2=i,b^a=be,c^a=cf,d^a=dei,c^b=cg,f^c=fh,g^c=gi

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, 228608);