#248921

order	512
nilpotency class	3
isoclinism	isoclinic class of #248899
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	C₂³×D₄	D₄
LCS	G ▷ C₂³×C₄ ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂⁴×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂² ▷ C₁

1566 conjugacy classes of subgroups; only the 188 normal subgroups are shown.

normal subgroups only

This group has 1566 conjugacy classes of subgroups — too many to draw. Only its 188 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)(7,11)(10,12)(13,14,17,20)(15,18,22,25)(16,19,23,26)(21,27,28,24);(1,2)(3,4)(13,15,17,22)(14,18,20,25)(16,21,23,28)(19,24,26,27);(2,4)(5,6,8,10)(7,12,11,9)(13,16)(14,19)(15,21)(17,23)(18,24)(20,26)(22,28)(25,27);(13,15)(14,18)(16,21)(17,22)(19,24)(20,25)(23,28)(26,27)

Polycyclic presentation:

2^9|a^2=h,b^2=h,c^2=i,f^2=i,b^a=be,c^a=cf,d^a=de,f^a=fi,c^b=cg,d^b=dh,f^b=fh,e^c=eh,f^c=fi,f^d=fh

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, 156117);