#134274

order	512
nilpotency class	3
isoclinism	isoclinic class of #7606
automorphism	?
homology	Abelianization (H₁) = C₂²×C₈ · Schur multiplier (H₂) = C₂³

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₂²×C₄	C₂×C₂²⋊C₄
commutator	C₂²×C₄	C₂²×C₈
Frattini, cosocle	C₂³×C₈	C₂³
socle	C₂³	C₂×C₂²⋊C₈
prime radical	C₂²×#41	C₂×C₄
LCS	G ▷ C₂²×C₄ ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂²×C₄ ◁ C₂²×C₄×C₈ ◁ G
LECS	G ▷ C₂³×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁

532 conjugacy classes of subgroups; only the 122 normal subgroups are shown.

normal subgroups only

This group has 532 conjugacy classes of subgroups — too many to draw. Only its 122 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) = 44
transitive degree	μ_t(G) = 256

A minimal faithful permutation representation — 3 generators on 44 points:

(2,4)(5,6)(7,11,12,9)(8,10)(13,14,19,25,20,26,29,21,17,23,30,38,18,24,15,27)(16,31,34,44,33,40,28,43,32,42,36,37,35,22,39,41);(1,2,3,4)(13,15,18,30,17,29,20,19)(14,21,24,25,23,27,26,38)(16,28,35,34,32,39,33,36)(22,37,42,43,40,44,31,41);(5,7)(6,9)(8,12)(10,11)(13,16)(14,22)(15,28)(17,32)(18,35)(19,36)(20,33)(21,37)(23,40)(24,42)(25,43)(26,31)(27,44)(29,39)(30,34)(38,41)

Polycyclic presentation:

2^9|a^2=f,b^2=dh,e^2=i,f^2=h,h^2=i,b^a=bd,c^a=ce,e^a=eg,e^c=ei,f^c=fgi

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, 41470);