#381040

order	512
nilpotency class	3
isoclinism	isoclinic class of #29742
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂×C₄³

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle, prime radical	C₂⁴	C₂²×D₄
commutator	C₂²×C₄	C₂³×C₄
Frattini, cosocle	C₂³×C₄	C₂⁴
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂⁴ ◁ C₂×C₄³ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

1246 conjugacy classes of subgroups; only the 370 normal subgroups are shown.

normal subgroups only

This group has 1246 conjugacy classes of subgroups — too many to draw. Only its 370 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) = 512

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

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

Polycyclic presentation:

2^9|a^2=h,b^2=eg,c^2=g,d^2=ei,f^2=i,h^2=i,b^a=be,c^a=cf,d^a=di,c^b=cg,d^c=de,f^c=fi,h^c=hi

Conjugacy classes & character table

80 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 288236);