#393848

order	512
nilpotency class	3
isoclinism	isoclinic class of #32884
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂³×C₄²

Aliases

No aliases for this group.

Subgroups

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

1354 conjugacy classes of subgroups; only the 426 normal subgroups are shown.

normal subgroups only

This group has 1354 conjugacy classes of subgroups — too many to draw. Only its 426 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:

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

Polycyclic presentation:

2^9|a^2=h,b^2=efh,c^2=f,d^2=g,e^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, 301044);