#380690

order	512
nilpotency class	3
isoclinism	isoclinic class of #29602
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂²×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×#376 ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

1318 conjugacy classes of subgroups; only the 338 normal subgroups are shown.

normal subgroups only

This group has 1318 conjugacy classes of subgroups — too many to draw. Only its 338 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) = 32
transitive degree	μ_t(G) = 512

A minimal faithful permutation representation — 4 generators on 32 points:

(5,6,7,8)(10,13)(14,16)(19,25)(20,27)(22,29)(24,31)(26,28)(30,32);(1,2,3,4)(9,10,12,13)(11,14,15,16)(17,18)(19,22)(20,24)(21,23)(25,29)(26,30)(27,31)(28,32);(2,4)(11,15)(14,16)(17,19,21,26)(18,22,23,30)(20,28,27,25)(24,32,31,29);(9,11,12,15)(10,14,13,16)(18,23)(22,30)(24,31)(29,32)

Polycyclic presentation:

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

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, 287886);