#346921

order	512
nilpotency class	3
isoclinism	isoclinic class of #346918
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁷

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	C₂×#120
commutator, Frattini, cosocle	C₂×C₄²	C₂⁴
prime radical	#25729	C₂
LCS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

2251 conjugacy classes of subgroups; only the 200 normal subgroups are shown.

normal subgroups only

This group has 2251 conjugacy classes of subgroups — too many to draw. Only its 200 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) = 256

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

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

Polycyclic presentation:

2^9|c^2=h,d^2=h,e^2=h,f^2=hi,b^a=be,c^a=cf,d^a=dg,e^a=eh,f^a=fhi,e^b=eh,f^b=fi,d^c=dh,e^c=ei,f^c=fhi

Conjugacy classes & character table

59 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 254117);