#393821

order	512
nilpotency class	3
isoclinism	isoclinic class of #32884
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂³×C₄³

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂⁴	C₂²×D₄
commutator	C₂²×C₄	C₂³×C₄
Frattini, cosocle	C₂³×C₄	C₂⁴
prime radical	C₂×#2657	C₂
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂⁴ ◁ C₂×#376 ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

2408 conjugacy classes of subgroups; only the 466 normal subgroups are shown.

normal subgroups only

This group has 2408 conjugacy classes of subgroups — too many to draw. Only its 466 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:

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

Polycyclic presentation:

2^9|a^2=h,b^2=ehi,c^2=f,d^2=g,e^2=i,h^2=i,b^a=be,c^a=cf,d^a=dg,e^b=ei,h^b=hi,d^c=di

Conjugacy classes & character table

104 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 301017);