#391867

order	512
nilpotency class	2
isoclinism	isoclinic class of #16249
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂⁶×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, Frattini, cosocle	C₂³×C₄	C₂⁴
commutator, socle	C₂⁴	C₂³×C₄
prime radical	C₂²×#127	C₄
LCS	G ▷ C₂⁴ ▷ C₁
UCS	C₁ ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

2078 conjugacy classes of subgroups; only the 322 normal subgroups are shown.

normal subgroups only

This group has 2078 conjugacy classes of subgroups — too many to draw. Only its 322 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:

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

Polycyclic presentation:

2^9|a^2=f,b^2=h,d^2=e,h^2=i,b^a=be,c^a=cf,c^b=cg,d^b=defi,d^c=de

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, 299063);