#380672

order	512
nilpotency class	3
isoclinism	isoclinic class of #29742
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂²×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	C₂³×D₄	C₂³
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂⁴ ◁ C₂×C₄³ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

1452 conjugacy classes of subgroups; only the 338 normal subgroups are shown.

normal subgroups only

This group has 1452 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) = 256

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

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

Polycyclic presentation:

2^9|a^2=h,b^2=egi,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, 287868);